Quick Sort (3 Way Partition)

Algorithm

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

# 3-way partition
i = 1, k = 1, p = n
while i < p,
  if a[i] < a[n], swap a[i++,k++]
  else if a[i] == a[n], swap a[i,--p]
  else i++
end
→ invariant: a[p..n] all equal
→ invariant: a[1..k-1] < a[p..n] < a[k..p-1]

# move pivots to center
m = min(p-k,n-p+1)
swap a[k..k+m-1,n-m+1..n]

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

Properties

• Not stable
• O(lg(n)) extra space
• O(n²) time, but typically O(n·lg(n)) time
• Adaptive: O(n) time when O(1) unique keys

Discussion

The 3-way partition variation of quick sort has slightly higher overhead compared to the standard 2-way partition version. Both have the same best, typical, and worst case time bounds, but this version is highly adaptive in the very common case of sorting with few unique keys.

The 3-way partitioning code shown above is written for clarity rather than optimal performance; it exhibits poor locality, and performs more swaps than necessary. A more efficient but more elaborate 3-way partitioning method is given in Quicksort is Optimal by Robert Sedgewick and Jon Bentley.

When stability is not required, quick sort is the general purpose sorting algorithm of choice. Recently, a novel dual-pivot variant of 3-way partitioning has been discovered that beats the single-pivot 3-way partitioning method both in theory and in practice.

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 p (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.

Dual Pivot Quicksort: Code and Discussion.

Bubble-sort with Hungarian ("Csángó") folk dance YouTube video, created at Sapientia University, Tirgu Mures (Marosvásárhely), Romania.

Select-sort with Gypsy folk dance YouTube video, created at Sapientia University, Tirgu Mures (Marosvásárhely), Romania.

Sorting Out Sorting, Ronald M. Baecker with the assistance of David Sherman, 30 minute color sound film, Dynamic Graphics Project, University of Toronto, 1981. Excerpted and reprinted in SIGGRAPH Video Review 7, 1983. Distributed by Morgan Kaufmann, Publishers. Excerpt.