Quick sort

Quick Sort
Afaq Mansoor Khan
BSSE 2017-21
IMSciences Peshawar

What is Quick Sort?
• Quick sort is a divide and conquer algorithm
which relies on a partition operation:
– to partition an array an element called a pivot is
selected
• All elements smaller than the pivot are moved
before it and all greater elements are moved
after it
• The lesser and greater sublists are then
recursively sorted
2

Quick sort
• Efficient implementations (with in-place
partitioning), somewhat complex, but are
among the fastest sorting algorithms in
practice
• One of the most popular sorting algorithms
and is available in many standard
programming libraries.

Idea of Quick-Sort
1) Divide : If the sequence S has 2 or more elements,
select an element x from S to be your pivot. Any
arbitrary element, like the last, will do. Remove all the
elements of S and divide them into 3 sequences:
L, holds S’s elements less than x
E, holds S’s elements equal to x
G, holds S’s elements greater than x
2) Recurse: Recursively sort L and G
3) Conquer: Finally, to put elements back into S in order,
first inserts the elements of L, then those of E, and
those of G.

Idea of Quick Sort
1) Select: pick an element
2) Divide: rearrange elements so
that x goes to its final position E
3) Recurse and Conquer:
recursively sort

Two key steps
• How to pick a pivot?
• How to partition?

Pick a pivot
• Use the first element as pivot
– if the input is random, ok
– if the input is presorted (or in reverse order)
• all the elements go into S2 (or S1)
• Results in O(n2) behavior (Analyze this case later)
• Choose the pivot randomly
– generally safe

A better partition
• Want to partition an array A[left .. right]
• First, get the pivot element out of the way by swapping it with the last
element. (Swap pivot and A[right])
pivot i j
5 6 4 6 3 12 19 5 6 4 63 1219
swap

– Move i right, skipping over elements smaller than the
pivot
– Move j left, skipping over elements greater than the pivot
i j
5 6 4 63 1219
i j
5 6 4 63 1219

• When i and j have stopped and i is to the left of j
– Swap A[i] and A[j]
• The large element is pushed to the right and the small element is pushed to
the left
– Repeat the process until i and j cross
swap
i j
5 6 4 63 1219
i j
5 3 4 66 1219

• When i and j have crossed
– Swap A[i] and pivot
• Result:
– A[x] <= pivot, for x < i
– A[x] >= pivot, for x > i
i j
5 3 4 66 1219
ij
5 3 4 66 1219
ij
5 3 4 6 6 12 19

Quicksort example
1
3
8
1
9
2
4
3
3
1
6
5
5
7
2
6
7
5
0
1
3
8
1
9
2
4
3
3
1
6
5
5
7
2
6
7
5
0
1
3 4
3
3
1
5
72
6
0
8
1
9
2
7
5
6
5
1
3
4
3
3
1
5
7
2
6
0
8
1
9
2
7
5
1
3
4
3
3
1
5
7
2
6
0
6
5
8
1
9
2
7
5
Select pivot
partition
Recursive call Recursive call
Merge

Quicksort Algorithm
• To sort a[left...right]:
1. if left < right:
1.1. Partition a[left...right] such that:
all a[left...p-1] are less than a[p], and
all a[p+1...right] are >= a[p]
1.2. Quicksort a[left...p-1]
1.3. Quicksort a[p+1...right]
2. Terminate

Quick Sort Source Code C++
void QuickSort(int list[], int left, int right) {
int pivot, leftArrow, rightArrow;
leftArrow = left;
rightArrow = right;
pivot = list[(left + right) / 2];
do {
while (list[rightArrow] > pivot)
rightArrow--;
while (list[leftArrow] < pivot)
leftArrow++;
if (leftArrow <= rightArrow) {
Swap_Data(list[leftArrow], list[rightArrow]);
leftArrow++;
rightArrow--;
}
}
while (rightArrow >= leftArrow);
if (left < rightArrow)
QuickSort(list, left, rightArrow);
if (leftArrow < right)
QuickSort(list, leftArrow, right);
}

Quick Sort Pseudo Code
• QuickSort(A,start,end)
• { if(start<end)
• { pIndex 🡨 partition(A,start,end)
QuickSort(A,start,Pindex-1)
QuickSort(A,Pindex+1,end)
}
}
• Partition(A,start,end)
• { pivot 🡨 A[end]
Pindex 🡨 start
For i 🡨 start to end-1
{
If(A[i] <= pivot)
{ Swap(a[i],P[pindex])
Pindex 🡨 Pindex+1
}
}
Swap(A[pindex] , A[end])
Return Pindex }

Complexity of Quick Sort
• Best case performance O(n log n)
• Average case performance O(n log n)
• Worst case performance O(n2)
• Worst case space complexity auxiliary O(log n)
• Where n is the number of elements to be sorted

References
• Data Structures and Algorithm Analysis by M. A.
Weiss, 2nd edition
• https://cs.nyu.edu/courses/spring99/V22.0102-
002/quick.html
• www.dcs.gla.ac.uk/~pat/52233/slides/QuickSort1x1.
pdf

Quick sort

Quick sort

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