Algorithms - "quicksort"

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

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.
2

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.
3

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.
4

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)}
5
A1 x ≤ v v A2 x ≥ v

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.
6

Quicksort - Example
Use 6 as the pivot
Sort left and right groups recursively
7
6 2 8 5 11 10 4 1 9 7 3
2 5 4 1 3 6 7 9 10 11 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.
8

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.
9

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
10

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.
11

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.
12

Partition Example
• A = {2, 8, 7, 1, 3, 5, 6, 4}
13

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.
14

• (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.
15

• (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.
16

• (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.
17

• (i) In lines 7–8, the pivot element is swapped so that it lies
between the two partitions.
18
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

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

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.
20

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.
21

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.
22
swap
pivot

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.
23

Partitioning Example Using Additional Array
Sort left and right groups recursively.
24
6 2 8 5 11 10 4 1 9 7 3a
b
2 85 11104 1 973 6

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.
25
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

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

In-Place Partitioning Example
27
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

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).
28

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).
29

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.
30

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)
31

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.
32

Algorithms - "quicksort"

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