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Few Unique Keys

Problem Size:  20 · 30 · 40 · 50     Magnification:  1x · 2x · 3x
Initial Condition:  Random · Nearly Sorted · Reversed · Few Unique


Sorting an array that consists of a small number of unique keys is common in practice. One would like an algorithm that adapts to O(n) time when the number of unique keys is O(1). In this example, there are 4 unique keys.

The traditional 2-way partitioning quicksort exhibits its worse-case O(n2) behavior here. For this reason, any quicksort implementation should use 3-way partitioning, where the array is partitioned into values less than, equal, and greater than the pivot. Because the pivot values need not be sorted recursively, 3-way quick sort adapts to O(n) time in this case.

Shell sort also adapts to few unique keys, though I do not know its time complexity in this case.


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  • Black values are sorted.
  • Gray values are unsorted.
  • A red triangle marks the algorithm position.
  • Dark gray values denote the current interval (shell, merge, quick).
  • A pair of red triangles marks the left and right pointers (quick).


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.