QuickSort
Jeff Chastine
Properties of QuickSort
• Runs in (n2) worst case, but (n lg n) on
average
• Constants in the (n lg n) are small, so it’s
often the best practical sorting algorithm
• Sorts in place
• Several variations:
– randomized partitioning
– random sampling
Jeff Chastine
Overview of QuickSort
• Divide: The array [p..r] is partitioned into two
non-empty subarrays A[p..q] and A[q+1 .. r]
such that each element in the first array is less
than or equal to each item in the second
• Conquer: Sort recursively
• Combine: Since they are already sorted
Jeff Chastine
QuickSort
QUICKSORT (A, p, r)
if p < r
then q ← PARTITION (A, p, r)
QUICKSORT (A, p, q)
QUICKSORT (A, q+1, r)
Note: obviously, PARTITION is the key here!
Jeff Chastine
Partitioning
• PARTITION selects a pivot element for
partitioning
• We have a range p to r in A with two variables
i and j, and a pivot point x, initialized to r.
• At the beginning of each loop:
– Everything between p and i is  x
– Everything between i and j is > x
Jeff Chastine
Jeff Chastine
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PARTITION
PARTITION (A, p, r)
1 x ← A[r]
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i← p-1
for j  p to r – 1
do if A[j]  x
then i ← i + 1
exchange A[i]  A[j]
exchange A[i + 1]  A[r]
return i + 1
Jeff Chastine
Performance
• Depends heavily on whether the partitioning
is balanced
• Unbalanced can run as slow as insertion sort!
• Worst-case: two subproblems with
– n – 1 elements
– 0 elements
• T (n) = T (n – 1) + T (0) + (n)
• Decomposes to (n2)
Jeff Chastine
What’s really going on...
• Average time not close to that...
• Suppose that we have a 9-to-1 split:
T (n) = T (9n/10) + T (n/10) + cn
• Then, a subproblem at depth i is 9in/10i
– Solving for i yields a tree of depth log10/9 n
• Thus, the height of the tree is still logarithmic!
• Even a 99-to-1 split is logarithmic
• Any split of constant proportionality yields a tree of
depth (n)
• Note: 80% of the time, PARTITION produces a split
more balanced than 9-to-1
Jeff Chastine
Randomized Quicksort
• Uses random sampling
• Use a randomly chosen element from the
array
• Swap A[r] with the chosen element
RANDOMIZED-PARTITION (A, p, r)
1 i  RANDOM (p, r)
2 exchange A[r]  A[i]
3 return PARTITION (A, p, r)
Jeff Chastine
Summary
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Runs in O(n log n) on average
Runs in O(n2) in worst-case
It uses partitioning to sort
Randomized partitioning helps
Quicksort is preferred, because of its low
constant factors
Jeff Chastine