Quicksort is the fastest known comparison-based sortingalgorithm (on average, and for a large number of elements),
requiring
steps. Quicksort is a recursive algorithm which first partitions an array according to several rules (Sedgewick 1978):

1. Some key
is in its final position in the array (i.e., if it is the th smallest, it is in position ).

2. All the elements to the left of are less than or equal to . The elements , ,
..., are called the "left subfile."

3. All the elements to the right of are greater than or equal to . The elements , ..., are called the "right subfile."

Quicksort was invented by Hoare (1961, 1962), has undergone extensive analysis and scrutiny (Sedgewick 1975, 1977, 1978), and is known to be about twice as fast as
the next fastest sorting algorithm. In the worst case,
however, quicksort is a slow algorithm (and for quicksort, "worst case" corresponds
to already sorted).

The average time
for the algorithm to sort a list of items arranged in random order is given by the recurrence
equation

(1)

with (Havil 2003, p. 129). This
recurrence can be rewritten as

(2)

which has solution

(3)

where is a harmonic
number. For ,
1, ..., the first few values are 0, 1, 3, 17/3, 53/6, 62/5, 163/10, ... (OEIS A093418 and A096620).

Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data
Structures and Algorithms. Reading, MA: Addison-Wesley, pp. 260-270,
1987.Havil, J. "Quicksort." §13.8 in Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 128-130,
2003.Hoare, C. A. R. "Partition: Algorithm 63,"
"Quicksort: Algorithm 64," and "Find: Algorithm 65." Comm.
ACM4, 321-322, 1961.Hoare, C. A. R. "Quicksort."
Computer J.5, 10-15, 1962.Knuth, D. E. The
Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed.
Reading, MA: Addison-Wesley, pp. 113-122, 1998.Press, W. H.;
Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Quicksort."
§8.2 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 323-327, 1992.Sedgewick, R. Quicksort.
Ph.D. thesis. Stanford Computer Science Report STAN-CS-75-492. Stanford, CA: Stanford
University, May 1975.Sedgewick, R. "The Analysis of Quicksort Programs."
Acta Informatica7, 327-355, 1977.Sedgewick, R. "Implementing
Quicksort Programs." Comm. ACM21, 847-857, 1978.Sloane,
N. J. A. Sequences A093418 and A096620 in "The On-Line Encyclopedia of Integer
Sequences."