Algorithms - "quicksort"

1. Quicksort
Dr. Ra’Fat A. AL-msie’deen
Chapter 4
Mutah University
Faculty of IT, Department of Software Engineering

2. Outlines: Quicksort
• Quicksort: Advantages and Disadvantages.
• Quicksort:
o Quicksort Function.
o Partition Function.
o Choice Of Pivot:
 Rightmost or Leftmost.
 Randomly.
 Median-of-Three Rule.
o Partitioning using Additional Array.
o In-Place Partitioning.
o Runtime of Quicksort (Worst, Best, Average).
• Randomized Quicksort.
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3. Quicksort
• Quicksort pros:
o Sorts in place. “rearrange the elements where they are”.
o Sorts θ(n lg n) in the average case.
o Very efficient in practice.
• Quicksort cons:
o Sorts θ(n2) in the worst case.
o not stable:
 does not preserve the relative order of elements with equal
keys.
 Sorting algorithm is stable if 2 records with the same key stay
in original order.
o But in practice, it’s quick.
o And the worst case doesn’t happen often … sorted.
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4. Quicksort
• Another divide-and-conquer algorithm:
• Divide: A[p…r] is partitioned (rearranged) into two nonempty
subarrays A[p…q-1] and A[q+1…r] such that each element of
A[p…q-1] is less than or equal to each element of A[q+1…r].
Index q is computed here, called pivot.
• Conquer: two subarrays are sorted by recursive calls to
quicksort.
• Combine: unlike merge sort, no work needed since the
subarrays are sorted in place already.
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5. Quicksort
• The basic algorithm to sort an array A consists of the following
four easy steps:
1. If the number of elements in A is 0 or 1, then return.
2. Pick any element v in A. This is called the pivot.
3. Partition A-{v} (the remaining elements in A) into two
disjoint groups:
o A1 = {x ∈ A-{v} | x ≤ v}, and
o A2 = {x ∈ A-{v} | x ≥ v}
4. return:
o {quicksort(A1) followed by v followed by quicksort(A2)}
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A1 x ≤ v v A2 x ≥ v

6. Quicksort
• Small instance has n ≤ 1.
o Every small instance is a sorted instance.
• To sort a large instance:
o select a pivot element from out of the n elements.
• Partition the n elements into 3 groups left, middle and right.
o The middle group contains only the pivot element.
o All elements in the left group are ≤ pivot.
o All elements in the right group are ≥ pivot.
• Sort left and right groups recursively.
• Answer is sorted left group, followed by middle group followed
by sorted right group.
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7. Quicksort - Example
Use 6 as the pivot
Sort left and right groups recursively
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6 2 8 5 11 10 4 1 9 7 3
2 5 4 1 3 6 7 9 10 11 8

8. Quicksort - Code
QUICKSORT(A, p, r)
1 if p < r
2 q = PARTITION(A, p, r)
3 QUICKSORT(A, p, q - 1)
4 QUICKSORT(A, q + 1, r)
o To sort an entire array A, the initial call is QUICKSORT(A, 1,
A:length).
o Initial call is Quicksort(A, 1, n), where n is the length of A.
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9. Partition
• Clearly, all the action takes place in the partition() function.
o Rearranges the subarray in place.
o End result:
 Two subarrays.
 All values in first subarray ≤ all values in second.
o Returns the index of the “pivot” element separating the
two subarrays.
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10. Partition - Code
• Partitioning the array: The key to the algorithm is the PARTITION
procedure, which rearranges the subarray A[p … r] in place.
PARTITION(A, p, r)
1 x = A[r]
2 i = p - 1
3 for j = p to r - 1
4 if A[j] ≤ x
5 i = i + 1
6 exchange A[i] with A[j]
7 exchange A[i + 1] with A[r]
8 return i + 1
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11. Partition - Code
• The four regions maintained by the procedure PARTITION on a
subarray A[p … r].
o The values in A[p ... i] are all less than or equal to x,
o the values in A[i + 1 ... j - 1] are all greater than x,
o and A[r] = x.
o The subarray A[j ... r - 1] can take on any values.
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12. Partition - Code
 The two cases for one
iteration of procedure
PARTITION.
1. (a) If A[j] > x, the only
action is to increment j ,
which maintains the loop
invariant.
2. (b) If A[j] ≤ x, index i is
incremented, A[i] and
A[j] are swapped, and
then j is incremented.
Again, the loop invariant
is maintained.
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13. Partition Example
• A = {2, 8, 7, 1, 3, 5, 6, 4}
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14. Partition Example - Explanation
• Lightly shaded elements are in the first partition with values ≤ x
(pivot).
• Heavily shaded elements are in the second partition with values ≥ x
(pivot).
• The unshaded elements have not yet been put in one of the first
two partitions.
• The final white element is the pivot.
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15. Partition Example - Explanation
• (a) The initial array and variable settings. None of the
elements have been placed in either of the first two
partitions.
• (b) The value 2 is “swapped with itself” and put in the
partition of smaller values.
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16. Partition Example - Explanation
• (c)–(d) The values 8 and 7 are added to the partition of larger
values.
• (e) The values 1 and 8 are swapped, and the smaller partition
grows.
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17. Partition Example - Explanation
• (f) The values 3 and 7 are swapped, and the smaller partition
grows.
• (g)–(h) The larger partition grows to include 5 and 6, and the
loop terminates.
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18. Partition Example - Explanation
• (i) In lines 7–8, the pivot element is swapped so that it lies
between the two partitions.
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PARTITION(A, p, r)
1 x = A[r]
2 i = p - 1
3 for j = p to r - 1
4 if A[j] ≤ x
5 i = i + 1
6 exchange A[i] with A[j]
7 exchange A[i + 1] with A[r]
8 return i + 1

19. Example: On an 8-element subarray.
19[The index j disappears because it is no longer needed once the for loop is exited.]

20. Choice Of Pivot
• Pivot is rightmost element in list that is to be sorted.
o When sorting A[6:20], use A[20] as the pivot.
o Textbook implementation does this.
• Randomly select one of the elements to be sorted as the
pivot.
o When sorting A[6:20], generate a random number r in the
range [6, 20].
o Use A[r] as the pivot.
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21. Choice Of Pivot
• Median-of-Three rule - from the leftmost, middle, and
rightmost elements of the list to be sorted, select the one
with median key as the pivot.
o When sorting A[6:20], examine A[6], A[13] ((6+20)/2), and
A[20].
o Select the element with median (i.e., middle) key.
o If A[6].key = 30, A[13].key = 2, and A[20].key = 10, A[20]
becomes the pivot.
o If A[6].key = 3, A[13].key = 2, and A[20].key = 10, A[6]
becomes the pivot.
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22. Choice Of Pivot
• If A[6].key = 30, A[13].key = 25, and A[20].key = 10, A[13]
becomes the pivot.
• When the pivot is picked at random or when the median-of-
three rule is used, we can use the quicksort code of the
textbook provided we first swap the rightmost element and
the chosen pivot.
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swap
pivot

23. Partitioning Into Three Groups
• Sort A = [6, 2, 8, 5, 11, 10, 4, 1, 9, 7, 3].
• Leftmost element (6) is the pivot.
• When another array B is available:
o Scan A from left to right (omit the pivot in this scan),
placing elements ≤ pivot at the left end of B and the
remaining elements at the right end of B.
o The pivot is placed at the remaining position of the B.
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24. Partitioning Example Using Additional Array
Sort left and right groups recursively.
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6 2 8 5 11 10 4 1 9 7 3a
b
2 85 11104 1 973 6

25. In-place Partitioning
1. Find leftmost element (bigElement) > pivot.
2. Find rightmost element (smallElement) < pivot.
3. Swap bigElement and smallElement provided
bigElement is to the left of smallElement.
4. Repeat.
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6 2 8 5 11 10 4 1 9 7 3a
6
6 2 3 5 11 10 4 1 9 7 8a
6
pivot

26. In-place Partitioning
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X : pivot
0 1 n-1
swap
the first element
greater than pivot
the first element
smaller than pivot

27. In-Place Partitioning Example
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6 2 8 5 11 10 4 1 9 7 3a 6 8 3
6 2 3 5 11 10 4 1 9 7 8a 6 11 1
6 2 3 5 1 10 4 11 9 7 8a 6 10 4
6 2 3 5 1 4 10 11 9 7 8a 6 104
bigElement is not to left of smallElement, terminate process.
Swap pivot and smallElement.
4 2 3 5 1 4 11 9 7 8a 6 106

28. Runtime of Quicksort
 Worst Case Partition:
o The running time of quicksort depends on whether the
partitioning is balanced or not.
• Best Case Partition:
o When the partitioning procedure produces two regions of
size n/2, we get the a balanced partition with best case
performance: T(n) = 2T(n/2) + Θ(n) = Θ(n lg n).
• Average Case Partition:
o Average complexity is also Θ(n lg n).
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29. Best Case Partitioning
• A recursion tree for quicksort in which partition always
balances the two sides of the partition equally (the best case).
The resulting running time is Θ(n lg n).
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30. Randomized Quicksort
• An algorithm is randomized if its behavior is determined not
only by the input but also by values produced by a random-
number generator.
• Exchange A[r] with an element chosen at random from A[p…r]
in Partition.
• This ensures that the pivot element is equally likely to be any
of input elements.
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31. Randomized Quicksort
RANDOMIZED-PARTITION(A, p, r)
1 i = RANDOM(p, r)
2 exchange A[r] with A[i]
3 return PARTITION(A, p, r)
RANDOMIZED-QUICKSORT(A, p, r)
1 if p < r
2 q = RANDOMIZED-PARTITION(A, p, r)
3 RANDOMIZED-QUICKSORT(A, p, q - 1)
4 RANDOMIZED-QUICKSORT(A, q + 1, r)
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32. Review: Analyzing Quicksort
• What will be the worst case for the algorithm?
o Partition is always unbalanced.
• What will be the best case for the algorithm?
o Partition is balanced.
• Will any particular input elicit the worst case?
o Yes: Already-sorted input.
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33. Quicksort
Dr. Ra’Fat A. AL-msie’deen
Chapter 4
Mutah University
Faculty of IT, Department of Software Engineering