Smart Sort

The Quicksort sorting algorithm has been around for 60 years, and, if implemented properly, it is still the fastest option for many sorting tasks. According to the description on Wikipedia, a well designed Quicksort is "…somewhat faster than Merge sort and about two or three times faster than Heapsort."

Many Linux users today have studied Quicksort at some point in the past, through a computer science class or other training scenario, but unless you are working as a professional programmer, chances are it has been a few years since you have taken the time to ponder the elegant Quicksort algorithm. Still, sorting goes on all the time on Linux networks. You don't have to be a full-time app developer to conjure up an occasional script to rank results or order a set of values extracted from a log file. This article explores some of the nuances of the classic Quicksort.

Quicksort ABC

The Quicksort [1] algorithm originated with Tony Hoare [2], who first developed it in 1959 and published it in 1961. Quicksort is what is known as a divide-and-conquer algorithm. One element in the array is chosen to be the pivot element. All elements smaller than the pivot element are then grouped in a sub-array before it, and all elements larger than the pivot element are placed in a sub-array after it. This process is then repeated with the sub-arrays: a pivot element is chosen, with smaller elements placed in a sub-array before and larger elements placed in a sub-array after. After a finite number of steps, the size of the sub-arrays becomes one, and at that point, the whole array has been sorted.

Too complicated? Figure 1 sums up the Quicksort algorithm. Boxes containing only one red number are already in the right position. In the first row, the number 6 is the pivot element. Now all of the elements are sorted in relation to 6. In the second row, the 5 and the 9 act as the new pivot elements. Now the partial arrays are sorted relative to 5 and 9. The result is the third row, where almost all of the elements are already sorted. Only the 8 (the new pivot element) has to swap places with the 7. If you would prefer an animated clip of the Quicksort algorithm, check out the Quicksort page on Wikipedia [1].

Figure 1: A schematic illustration of the Quicksort algorithm.

If the array contains n elements, an average of n*log(n) sorting steps are needed. The log(n) factor results from the fact that the algorithm halves the array in each step. In a worst-case scenario, Quicksort requires n*n sorting steps. To illustrate this worst-case scenario, consider that, in Figure 1, the middle element served as the pivot element. But any other element could also serve as the pivot element. If the first element is the pivot element and the array is already sorted in ascending order, the array must be halved exactly n times, which would be the (very unlikely) worst case. Note that a few lines of text are all I needed to describe the elegant and highly efficient Quicksort algorithm, along with its performance characteristics.

First Encounter

The classic Quicksort implementation in C lacks charm and is quite successful at disguising its elegant design, as Listing 1 demonstrates. I won't provide a full description of the code; however, one observation is very interesting.

01 void quickSort(int arr[], int left, int right) {
02 int i = left, j = right;
03 int tmp;
04 int pivot = arr[abs((left + right) / 2)];
05 while (i <= j) {
06   while (arr[i] < pivot) i++;
07     while (arr[j] > pivot) j--;
08     if (i <= j) {
09       tmp = arr[i];
10       arr[i] = arr[j];
11       arr[j] = tmp;
12       i++; j--;
13     }
14   }
15   if (left < j) quickSort(arr, left, j);
16   if (i < right) quickSort(arr, i, right);
17 }

In Lines 9 to 11, the code overwrites the existing elements. Thus, the algorithm runs in-place and assumes mutable data. A nice expression has been established for the task of overwriting old values with new ones in functional programming: destructive assignment [3]. This takes us neatly to the next topic: in functional programming languages like Haskell, Quicksort is represented in a far more elegant way.

Second Encounter

In Haskell, data is immutable, which precludes destructive assignment by design. The Haskell-based Quicksort algorithm in Listing 2 creates a new list in each iteration, rather than acting directly on the array as in Listing 1. Quicksort in two lines? Is that all there is to it? Yes.

qsort [] = []
qsort (x:xs) = qsort [y | y <- xs, y < x] ++ [x] ++ qsort [y | y <- xs, y >= x]

The qsort algorithm consists of two function definitions. The first line applies the defined Quicksort to the empty list. The second line represents the general case, where the list consists of at least one element: x:xs. Here, by convention, x denotes the beginning of the list and xs denotes the remainder.

The strategy of the Quicksort algorithm can be implemented almost directly in Haskell:

• use the first element of the list x as the pivot element;
• insert ((++)) all elements in xs that are lesser than x ((qsort [y | y <- xs, y < x])) in front of the one-element list [x];
• append all elements in xs that are at least as large as x to the list [x] ((qsort [y | y <- xs, y >= x])).

The recursion ends when Quicksort is applied to the empty list. Admittedly, the compactness of Haskell seems unusual. However, this Quicksort algorithm can be implemented in any programming language that supports list comprehension – which leads to the more mainstream Python programming language in Listing 3.

def qsort(L):
  if len(L) <= 1: return L
  return qsort([lt for lt in L[1:]   if lt < L[0]]) + L[0:1] +   qsort([ge for ge in L[1:] if ge >= L[0]])

The description of the Haskell algorithm can be applied almost verbatim to Python. The subtle difference is that in Python, L[0:1] acts as the first element and you express the list concatenation in Python with the + symbol. The algorithm reliably performs its services in Figure 2.

Figure 2: Using the Quicksort algorithm in Python.