Unit vii sorting

Unit – VII
Sorting
Prepared By:
Dabbal Singh Mahara
1

• Introduction
• Bubble sort
• Insertion sort
• Selection sort
• Quick sort
• Merge sort
• Comparison and efficiency of sorting
Contents
2

 Sorting is among the most basic problems in algorithm design.
 Sorting is important because it is often the first step in more complex
algorithms.
 Sorting is to take an unordered set of comparable items and arrange them
in some order.
 That is, sorting is a process of arranging the items in a list in some order
that is either ascending or descending order.
 Let a[n] be an array of n elements a0,a1,a2,a3........,an-1 in memory. The
sorting of the array a[n] means arranging the content of a[n] in either
increasing or decreasing order.
i.e. a0<=a1<=a2<=a3<.=.......<=an-1
Introduction
3
• Efficient sorting is important for optimizing the use of other algorithms
(such as search and merge algorithms) that require sorted lists to work
correctly.

Terminology
● Internal Sort:
Internal sorting algorithms assume that data is stored in an array in main memory
of computer. These methods are applied to small collection of data. That is, the
entire collection of data to be sorted is small enough that the sorting can take place
within main memory. Examples are: Bubble, Insertion, Selection, Quick, merge
etc.
• External Sort:
When collection of records is too large to fit in the main memory, records must
reside in peripheral or external memory. The only practical way to sort it is to read
some records from the disk do some rearranging then write back to disk. This
process is repeated until the file is sorted. The sorting techniques to deal with
these problems are called external sorting. Sorting large collection of records is
central to many applications, such as processing of payrolls and other business
databases. . Example: external merge sort
4

• In-place Sort
The algorithm uses no additional array storage, and hence (other than
perhaps the system’s recursion stack) it is possible to sort very large lists
without the need to allocate additional working storage.
Examples are: Bubble sort, Insertion Sort, Selection sort
• Stable Sort:
sort is said to be stable if elements with equal keys in the input list are
kept in the same order in the output list.
If all keys are different then this distinction is not necessary. But if there are
equal keys, then a sorting algorithm is stable if whenever there are two records
(let's say R and S) with the same key, and R appears before S in the original list,
then R will always appear before S in the sorted list.
 However, assume that the following pairs of numbers are to be sorted by their
first component:
• (4, 2) (3, 7) (3, 1) (5, 6)
• (3, 7) (3, 1) (4, 2) (5, 6) (order maintained)
• (3, 1) (3, 7) (4, 2) (5, 6) (order changed)
• Adaptation to Input:
if the sorting algorithm takes advantage of the sorted or nearly sorted
input, then the algorithm is called adaptive otherwise not. Example:
insertion sort is adaptive 5

512354277 101
1 2 3 4 5 6
5 12 35 42 77 101
1 2 3 4 5 6
• Input: A sequence of n numbers a1, a2, . . . , an
• Output: A permutation (reordering) a1’, a2’, . . . , an’ of the input
sequence such that a1’ ≤ a2’ ≤ · · · ≤ an’
The Sorting Problem
6

Elementary Sorting methods
Bubble Sort:
• The basic idea of this sort is to pass through the array
sequentially several times.
• Each pass consists of comparing each element in the
array with its successor (for example a[i] with a[i + 1])
and interchanging the two elements if they are not in the
proper order.
• After each pass an element is placed in its proper place
and is not considered in succeeding passes.
7

"Bubbling Up" the Largest Element
• Traverse a collection of elements
– Move from the front to the end
– “Bubble” the largest value to the end using pair-
wise comparisons and swapping
512354277 101
1 2 3 4 5 6
8

512354277 101
1 2 3 4 5 6
Swap42 77
9

512357742 101
1 2 3 4 5 6
Swap35 77
10

512773542 101
1 2 3 4 5 6
Swap12 77
11

577123542 101
1 2 3 4 5 6
No need to swap
12

577123542 101
1 2 3 4 5 6
Swap5 101
13

77123542 5
1 2 3 4 5 6
101
Largest value correctly placed
14

Items of Interest
• Notice that only the largest value is correctly
placed
• All other values are still out of order
• So we need to repeat this process
77123542 5
1 2 3 4 5 6
101
Largest value correctly placed
15

Repeat “Bubble Up” How Many
Times?
• If we have N elements…
• And if each time we bubble an element, we
place it in its correct location…
• Then we repeat the “bubble up” process N –
1 times.
• This guarantees we’ll correctly
place all N elements.
16

“Bubbling” All the Elements
77123542 5
1 2 3 4 5 6
101
5421235 77
1 2 3 4 5 6
101
4253512 77
1 2 3 4 5 6
101
4235512 77
1 2 3 4 5 6
101
4235125 77
1 2 3 4 5 6
101
N-1

Reducing the Number of Comparisons
12354277 101
1 2 3 4 5 6
5
77123542 5
1 2 3 4 5 6
101
5421235 77
1 2 3 4 5 6
101
4253512 77
1 2 3 4 5 6
101
4235512 77
1 2 3 4 5 6
101

Algorithm
BubbleSort(A, n)
{
for(i = 0; i <n-1; i++)
{
for(j = 0; j < n-i-1; j++)
{
if(A[j] > A[j+1])
{
temp = A[j];
A[j] = A[j+1];
A[j+1] = temp;
}
}
}
}
19

Properties:
 Stable
 O(1) extra space
 O(n2) comparisons and swaps
 Adaptive: O(n) when nearly sorted input
 Higher overhead than insertion sort
Time Complexity:
Inner loop executes for (n-1) times when i=0, (n-2) times when i=1 and
so on:
Time complexity = (n-1) + (n-2) + (n-3) + …………………………. +2 +1
= O(n2)
Space Complexity:
Since no extra space besides 3 variables is needed for sorting
Space complexity = O(n)
20

Here, we notice that after each pass, an element is placed in its proper order
and is not considered succeeding passes. Furthermore, we need n-1 passes
to sort n elements. 21
Example:

Selection Sort
Idea: Find the least (or greatest) value in the array, swap it into the leftmost(or
rightmost)component (where it belongs), and then forget the leftmost
component. Do this repeatedly.
Let a[n] be a linear array of n elements. The selection sort works as
follows:
pass 1:
Find the location loc of the smallest element in the list of n
elements a[0], a[1], a[2], a[3], …......,a[n-1] and then interchange
a[loc] and a[0].
Pass 2:
Find the location loc of the smallest element in the sub-list of n-1
elements a[1], a[2],a[3], …......,a[n-1] and then interchange a[loc]
and a[1] such that a[0], a[1] ............. and so on.
Finally, we will get the sorted list a[0]<=a[1]<= a[2]<=a[3]<= .....<= a[n-1].
22
Selection Sort Example

Algorithm:
SelectionSort(A)
{
for( i = 0;i < n-1 ;i++)
{
least=A[i];
loc=i;
for ( j = i + 1;j < n ;j++)
{
if (A[j] < least)
least= A[j];
loc=j;
}
if(i!=loc)
swap(A[i],A[loc]);
}
}
23

24
Example:
Consider the array: 15, 10, 20, 25, 5
After Pass 1: 5, 10, 20, 25, 15
After Pass 2: 5, 10, 20, 25, 15
After Pass 3: 5, 10, 15, 25, 20
After Pass 4: 5, 10, 15, 20, 25
Time Complexity:
Inner loop executes for (n-1) times when i=0, (n-2) times when i=1 and
so on: Time complexity = (n-1) + (n-2) + (n-3) + …………………………. +2 +1
= O(n2)
The complexity of this algorithm is same as that of bubble sort, but
number of swap operations is reduced greatly.

25
Properties:
 Not- stable
 In-place sorting (O(1) extra space)
 Most time depends upon comparisons O(n2) comparisons
 Not adaptive
 Minimum number of swaps, so in the applications where
cost of swapping items is high selection sort is the
algorithm of choice.
Space Complexity:
Since no extra space besides 5 variables is needed for sorting,

Insertion Sort
Idea:
like sorting a hand of playing cards start with an empty left hand and the
cards facing down on the table.
Remove one card at a time from the table,
Compare it with each of the cards already in the hand, from right to left
and insert it into the correct position in the left hand. The cards held in
the left hand are sorted
Suppose an array a[n] with n elements. The insertion sort works as follows:
pass 1:
a[0] by itself is trivially sorted.
Pass 2:
a[1] is inserted either before or after a[0] so that a[0], a[1] is sorted.
Pass 3:
a[2] is inserted into its proper place in a[0],a[1] that is before a[0],
between a[0] and a[1], or after a[1] so that a[0],a[1],a[2] is sorted.
.......................................
pass N:
a[n-1] is inserted into its proper place in a[0],a[1],a[2],........,a[n-2] so that
a[0],a[1],a[2],............,a[n-1] is sorted with n elements.
26

28
Algorithm:
InsertionSort(A)
{
for( i = 1;i < n ;i++)
{
temp = A[i]
for ( j = i -1;j >=0;j--)
{
if(A[j]>temp )
A[j+1] = A[j];
}
A[j+1] = temp;
}
}

30
Best case:
If array elements are already sorted, inner loop executes
only 1 time for i=1,2,3,… , n-1 for each. So,
total time complexity = 1+1+1+ …………..+1 (n-1)
times = n-1 = O(n)
Space Complexity:
Since no extra space besides 3variables is needed for
sorting,

31
Properties:
 Stable Sorting
 In-place sorting (O(1) extra space)
 Most time depends upon comparisons O(n2)
comparisons
 Run time depends upon input (O(n) when nearly sorted
input)
 O(n2) comparisons and swaps

32
• This algorithm is based on the divide and conquer paradigm.
• The main idea behind this sorting is: partitioning of the elements into
two groups and sort these parts recursively. The two partitions contain
values that are either greater or smaller than the key .
• It possesses very good average case complexity among all the sorting
algorithms.
Quick-Sort
Steps for Quick Sort:
Divide: partition the array into two non-empty sub arrays.
Conquer: two sub arrays are sorted recursively.
Combine: two sub arrays are already sorted in place so no need to combine

33
Quicksort
• 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

34
Partitioning in Quicksort
• A key step in the Quicksort algorithm is partitioning the
array
– We choose some (any) number p in the array to use as a pivot
– We partition the array into three parts:
p
numbers less
than p
numbers greater than or
equal to p
p

35
Partitioning in Quicksort
• Choose an array value (say, the first) to use as the
pivot
• Starting from the left end, find the first element
that is greater than the pivot
• Searching backward from the right end, find the
first element that is less than or equal to the pivot
• Interchange (swap) these two elements
• Repeat, searching from where we left off, until
done

36
Partitioning
• To partition a[left...right]:
1. Set pivot = a[left], l = left + 1, r = right;
2. while l < r, do
2.1. while l < right & a[l] <= pivot , set l = l + 1
2.2. while r > left & a[r] > pivot , set r = r - 1
2.3. if l < r, swap a[l] and a[r]
3. Set a[left] = a[r], a[r] = pivot
4. Terminate

37
Algorithm:
QuickSort(A,l,r)
{
if(l<r)
{
p = Partition(A,l,r);
QuickSort(A,l,p-1);
QuickSort(A,p+1,r);
}
}
Properties:
• Not -stable Sorting
 In-place sorting (O(log n) extra space)
 Not Adaptive
 O(n2) time, typically O(nlogn)

38
Partition(A,l,r)
{
x =l;
y =r ;
p = A[l];
while(x<y)
{
while(A[x] <= p)
x++;
while(A[y] >p)
y--;
if(x<y)
swap(A[x],A[y]);
}
A[l] = A[y];
A[y] = p;
return y; //return position of pivot
}

39
Example: a[]={5, 3, 2, 6, 4, 1, 3, 7}
(1 3 2 3 4) 5 (6 7)
and continue this process for each sub-arrays and finally we get a sorted array.

6 5 9 12 3 4
0 1 2 3 4 5
quickSort(arr,0,5)

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
pivot= ?
Partition Initialization...

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
pivot=6

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6
right moves to the left until
value that should be to left
of pivot...

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6

quickSort(arr,0,5)
6 5 9 12 3 4
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6
left moves to the right until
value that should be to right
of pivot...

quickSort(arr,0,5)
6 5 4 12 3 9
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6
of pivot...

quickSort(arr,0,5)
6 5 4 12 3 9
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6

quickSort(arr,0,5)
6 5 4 12 3 9
0 1 2 3 4 5
partition(arr,0,5)
left right
pivot=6
swap arr[left] and arr[right]

quickSort(arr,0,5)
6 5 4 3 12 9
0 1 2 3 4 5
partition(arr,0,5)
left
right
pivot=6
right & left CROSS!!!

quickSort(arr,0,5)
6 5 4 3 12 9
0 1 2 3 4 5
partition(arr,0,5)
left
right
pivot=6
1 - Swap pivot and arr[right]

quickSort(arr,0,5)
3 5 4 6 12 9
0 1 2 3 4 5
partition(arr,0,5)
left
right
pivot=6

quickSort(arr,0,5)
3 5 4 6 12 9
0 1 2 3 4 5
partition(arr,0,5)
left
right
pivot=6
2 - Return new location of pivot to caller
return 3

quickSort(arr,0,5) 3 5 4 6 12 9
0 1 2 3 4 5
Recursive calls to quickSort()
using partitioned array...
pivot
position

quickSort(arr,0,5)
3 5 4 6 12 9
0 1 2 3 4 5
quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5

quickSort(arr,0,5)
quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left right
quickSort(arr,0,5)

of pivot...
quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left right
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left right
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left
right
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
partition(arr,0,3)
left
right
2 - Return new location of pivot to caller
return 0
quickSort(arr,0,5)

quickSort(arr,0,3)
3 5 4 6
0 1 2 3
quickSort(arr,4,5)
12 9
4 5
quickSort(arr,0,5)
Recursive calls to quickSort()
using partitioned array...

quickSort(arr,0,3) quickSort(arr,4,5)
12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
Base case triggered...
halting recursion.

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
Base Case: Return

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left right
of pivot...

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left right

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left right
of pivot...

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left
right

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left
right
right & left CROSS!

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
5 4 6
1 2 3
partition(arr,1,3)
left
right
right & left CROSS!
1- swap pivot and arr[right]

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
4 5 6
1 2 3
partition(arr,1,3)
left
right
right & left CROSS!

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
4 5 6
1 2 3
partition(arr,1,3)
left
right
right & left CROSS!
2 – return new position of pivot
return 2

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
4 5 6
1 2 3

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
4 5 6
1 2 3
partition(arr,1,2)

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2

12 9
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2
partition(arr,4,5)

9 12
4 5
quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2
partition(arr,4,5)

quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2
quickSort(arr,4,5)
9 12
4 5
quickSort(arr,6,5)

quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2
quickSort(arr,4,5)
9 12
4 5
partition(arr,4,5)
quickSort(arr,6,5)

quickSort(arr,0,5)
quickSort(arr,0,0)
3
0
quickSort(arr,1,3)
6
3
4 5
1 2
quickSort(arr,4,5)
9
12
4
5
quickSort(arr,4,4)
quickSort(arr,5,5)
quickSort(arr,6,5)

97
• Best Case:
Divides the array into two partitions of equal size, therefore
T(n) = 2T(n/2) + O(n) , Solving this recurrence we get,
T(n)=O(nlogn)
• Worst case:
when one partition contains the n-1 elements and another partition contains
only one element.Therefore its recurrence relation is:
T(n) = T(n-1) + O(n), Solving this recurrence we get, T(n)=O(n2)
• Average case:
Good and bad splits are randomly distributed across throughout the tree
T1(n)= 2T'(n/2) + O(n) Balanced
T'(n)= T(n –1) + O(n) Unbalanced
Solving:
B(n)= 2(B(n/2 –1) + Θ(n/2)) + Θ(n)
= 2B(n/2 –1) + Θ(n)
= O(nlogn)
=>T(n)=O(nlogn)
Analysis of Quick Sort

98
Merge Sort
Merge sort is divide and conquer algorithm. It is recursive algorithm
having three steps. To sort an array A[l . . r]:
• Divide
Divide the n-element sequence to be sorted into two sub-sequences of
n/2 elements
• Conquer
Sort the sub-sequences recursively using merge sort. When the size of
the sequences is 1 there is nothing more to do
•Combine
Merge the two sorted sub-sequences into single sorted array. Keep
track of smallest element in each sorted half and inset smallest of two
elements into auxiliary array. Repeat this until done.

99
Algorithm:
MergeSort(A, l, r)
{
If(l<r)
{
m=(l+r)/2
MergeSort(A, l, m)
MergeSort(A, m+1, r)
Merge(A, l, m+1, r)
}
}
Properties:
 stable Sorting
 Not In-place sorting (O( n)
extra space)
 Not Adaptive
 Time complexity: O(nlogn)

100
Merge(A, B, l, m, r)
{
x= l, y=m, k =l;
While(x<m && y<r)
{
if(A[x] < A[y])
{
B[k] = A[x];
k++; x++;
}
else
{
B[k] = A[y];
k++; y++;
}
}
while(x<m)
{
B[k] = A[x];
k++;x++;
}
while(y<r)
{
B[k] = A[y];
k++; y++;
}
For(i =1;i<=r;i++)
{
A[i] = B[i];
}
}

101
Example: a[]= {4, 7, 2, 6, 1, 4, 7, 3, 5, 2, 6}

103
Time Complexity:
Recurrence Relation for Merge sort:
T(n) = 1 if n=1
T(n) = 2 T(n/2) + O(n) if n>1
Solving this recurrence we get
T(n) = O(nlogn)
Space Complexity:
It uses one extra array and some extra variables during sorting,
therefore,
Space Complexity = 2n + c = O(n)

Unit vii sorting

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Unit vii sorting