Quick Sort

Algorithm

# choose pivot
swap a[1,rand(1,n)]

# 2-way partition
k = 1
for i = 2:n, if a[i] < a[1], swap a[++k,i]
swap a[1,k]
→ invariant: a[1..k-1] < a[k] <= a[k+1..n]

# recursive sorts
sort a[1..k-1]
sort a[k+1,n]

Properties

• Not stable
• O(lg(n)) extra space (see discussion)
• O(n²) time, but typically O(n·lg(n)) time
• Not adaptive

Discussion

When carefully implemented, quick sort is robust and has low overhead. When a stable sort is not needed, quick sort is an excellent general-purpose sort -- although the 3-way partitioning version should always be used instead.

The 2-way partitioning code shown above is written for clarity rather than optimal performance; it exhibits poor locality, and, critically, exhibits O(n²) time when there are few unique keys. A more efficient and robust 2-way partitioning method is given in Quicksort is Optimal by Robert Sedgewick and Jon Bentley. The robust partitioning produces balanced recursion when there are many values equal to the pivot, yielding probabilistic guarantees of O(n·lg(n)) time and O(lg(n)) space for all inputs.

With both sub-sorts performed recursively, quick sort requires O(n) extra space for the recursion stack in the worst case when recursion is not balanced. This is exceedingly unlikely to occur, but it can be avoided by sorting the smaller sub-array recursively first; the second sub-array sort is a tail recursive call, which may be done with iteration instead. With this optimization, the algorithm uses O(lg(n)) extra space in the worst case.

Directions

• Click on above to restart the animations in a row, a column, or the entire table.
• Click directly on an animation image to start or restart it.
• Click on a problem size number to reset all animations.

Key

• Black values are sorted.
• Gray values are unsorted.
• Dark gray values denote the current interval.
• A pair of red triangles mark k and i (see the code).

References

Algorithms in Java, Parts 1-4, 3rd edition by Robert Sedgewick. Addison Wesley, 2003.

Programming Pearls by Jon Bentley. Addison Wesley, 1986.

Quicksort is Optimal by Robert Sedgewick and Jon Bentley, Knuthfest, Stanford University, January, 2002.