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AD8251 - DATA STRUCTURES DESIGN
Unit III: SORTING AND SEARCHING

Unit III: Sorting and Searching
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• Bubble sort – selection sort – insertion sort – merge sort –
quick sort – linear search – binary search – hashing – hash
functions – collision handling – load factors, rehashing,
and efficiency

• Sorting is an operation that segregates items into groups
according to specified criterion.
• Arranging items of the same kind, class or nature in some
ordered sequence.
• A = { 3 1 6 2 1 3 4 5 9 0 } A = { 0 1 1 2 3 3 4 5 6 9 }
• Sorting is a fundamental application for any domain
• Sorting help done to make searching faster
Sorting: Introduction
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Examples:
• Words in a dictionary are sorted
• Files in a directory are often listed in sorted order.
• The index of a book is sorted
• Sorting Books in Library
• Sorting Individuals by Height (Feet and Inches)
• Sorting Movies in Blockbuster (Alphabetical)
• Sorting Numbers (Sequential)
• etc
Sorting: Introduction
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• Imagine finding the phone number of your friend in
your mobile phone, but the phone book is not sorted.
• Need
– Efficient and quick data access
– Efficient data manipulation
Sorting: Introduction
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• A sorting algorithm is an algorithm that puts elements of a list
in a certain order.
– The most used orders
lexicographical order.
• Types
– Bubble Sort
– Selection Sort
– Insertion Sort
– Quick Sort
– Merge Sort
– etc.,
are numerical order and
–
Sorting Algorithm
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Selection Sort
• Selection sort is a simple sorting algorithm. It selects the
smallest element from an unsorted list in each iteration and
places that element at the beginning of the unsorted list.
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Selection Sort
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Algorithm
Step 1: Start with the 1st element, scan the entire list to find its
smallest element and exchange it with the 1st element.
Step 2: Start with the 2nd element, scan the remaining list to find
the smallest among the last (N-1) elements and exchange it with
the 2nd element.
Step 3: Continue this until the entire list is sorted.

Selection Sort: Example
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Selection Sort: Implementation
def selectionsort(t): #Method I:
for i in range(len(t)):
min_index=i
for j in range(i+1,len(t)):
if t[j] < t[min_index]:
min_index=j
t[min_index],t[i]=t[i],t[min_index]
print("List elements after",i+1,"phase is",t)
def selectionsort(t): #Method II:
for i in range(len(t)):
smallest=min(t[i:])
index_of_smallest=t.index(smallest,i)
t[i],t[index_of_smallest]=t[index_of_smallest],t[i]
print("List elements after",i+1,"phase is",t)
t=list(map(int,input("Enter the List elements separated by
space:").split(" ")))
print("List before sorting is",t)
selectionsort(t)
print("List after sorting is",t)

Selection Sort: Implementation
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Output:
Enter the List elements separated by space:6 2 19 3 2
List before sorting is [6, 2, 19, 3, 2]
List elements after 1 phase is [2, 6, 19, 3, 2]
List elements after 2 phase is [2, 2, 19, 3, 6]
List elements after 3 phase is [2, 2, 3, 19, 6]
List elements after 4 phase is [2, 2, 3, 6, 19]
List elements after 5 phase is [2, 2, 3, 6, 19]
List after sorting is [2, 2, 3, 6, 19]

• Sorting a list by inserting element at appropriate position.
• An example of an insertion sort occurs in everyday life while
playing cards.
– To sort the cards in your hand you extract a card, shift the
remaining cards, and then insert the extracted card in the
correct place.
– This process is repeated until all the cards are in the correct
sequence.
Insertion Sort
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To insert 12, we need to
make room for it by moving
first 36 and then 24.
Insertion Sort
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Insertion Sort
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Insertion Sort
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Insertion Sort
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Algorithm
Step 1: In every pass, an element is compared with all its previous
elements.
Step 2: If at some point it is found that the element can be inserted
at a position then space is created by shifting other elements one
position to the right and insert the element at the suitable
position.
Step 3: The steps are repeated for all elements in the array.
Insertion Sort
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Insertion Sort: Example

Insertion Sort: Implementation
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def insertionsort(t):
for i in range(1,len(t)):
key=t[i]
j=i-1
while j>=0 and t[j]>key:
t[j+1]=t[j]
j=j-1
t[j+1]=key
print("List elements after",i,"phase is",t)
t=list(map(int,input("Enter the List elements separated by
space:").split(" ")))
print("List before sorting is",t)
insertionsort(t)
print("List after sorting is",t)

Insertion Sort: Implementation
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Output:
Enter the List elements separated by space:6 2 19 4 2
List before sorting is [6, 2, 19, 4, 2]
List elements after 1 phase is [2, 6, 19, 4, 2]
List elements after 2 phase is [2, 6, 19, 4, 2]
List elements after 3 phase is [2, 4, 6, 19, 2]
List elements after 4 phase is [2, 2, 4, 6, 19]
List after sorting is [2, 2, 4, 6, 19]

Bubble Sort
• Bubble Sort is the simplest sorting algorithm that works by
repeatedly swapping the adjacent elements if they are in
wrong order.
• It is also known as exchange sort.
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Algorithm
• Compare adjacent elements and exchange them if they are out of
order
Step I: Comparing the first two elements, the second and third
elements, and so on, will move the largest (or smallest)
elements to the end of the array
Step II: Repeating this process will eventually sort the array
into ascending (or descending) order
• Bubble up - After each pass the largest element is bubbled to its
position.
Bubble Sort
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7 2 8 5 4
2 7 8 5 4
8>5 (T)
2 7 8 5 4
2 7 5 8 4
2 7 5 4 8
2 5 7 4 8
2 5 4 7 8
2 7 5 4 8
2 5 4 7 8
2 4 5 7 8
2 5 4 7 8
2 4 5 7 8
2 4 5 7 8
All elements
Sorted
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Bubble Sort: Example
2>7 (F) No Swap 2>5 (F) No Swap
7>2 (T) Swap
7>8 (F) No Swa
2 7 5 4 8
p
Swap
8>4 (T) Swap
7>5 (T) Swap
7>4 (T) Swap
5>4 (T) Swap
2>4 (F) No Swap
Iteration I

Bubble Sort: Implementation
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def bubblesort(t):
n=len(t)
for i in range(n):
for j in range(0,n-i-1):
if t[j] > t[j+1]:
t[j],t[j+1]=t[j+1],t[j] #Swap
print("List elements after",i+1,"phase is",t)
t=list(map(int,input("Enter the List elements separated by
space:").split(" ")))
print("List before sorting is",t)
bubblesort(t)
print("List after sorting is",t)

Bubble Sort: Implementation
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Output:
Enter the List elements separated by space:12 2 3 11 1 10
List before sorting is [12, 2, 3, 11, 1, 10]
List elements after 1 phase is [2, 3, 11, 1, 10, 12]
List elements after 2 phase is [2, 3, 1, 10, 11, 12]
List elements after 3 phase is [2, 1, 3, 10, 11, 12]
List elements after 4 phase is [1, 2, 3, 10, 11, 12]
List elements after 5 phase is [1, 2, 3, 10, 11, 12]
List elements after 6 phase is [1, 2, 3, 10, 11, 12]
List after sorting is [1, 2, 3, 10, 11, 12]

• Sort a sequence of n elements into non-decreasing order. This
algorithm is an example of divide and conquer approach.
– Divide: Divide the n-element sequence to be sorted into two
subsequences of n/2 elements each
– Conquer: Sort the two subsequences recursively using merge sort.
– Combine:
answer.
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Merge Sort
sub
problem 2
of size n/2
sub
problem 1
of size n/2
a solution to
sub problem 1
a solution to
the original
problem
a solution to
sub problem 2
Merge the two sorted subsequences to produce the sorted
a problem
of size n

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Merge Sort: Example

Algorithm
• Split list into about equal halves and make copies of eac2h8 half in lists
B and C
• Sort arrays B and C recursively
• Merge sorted arrays B and C as follows:
• Repeat the following until no elements remain in one of the list:
• compare the first elements in the remaining unprocessed
portions of the list
• copy the smaller of the two into original list, while incrementing
the index indicating the unprocessed portion of that list
• Once all elements in one of the lists are processed, copy the
remaining unprocessed elements from the other list into original
Merge Sort

3 27 38 43 10 82
R 9
L
0 1 2 3 0 1 2
i j
A
0 1 2 3 4 5
k
Initially, i=0,
j=0, k=0
L[i]<
6
= R[j]
A[k]=L[i]
++i, ++k
3 27 38 43 10 82
R 9
L
i j
3
0
A
1 2 3 4 5
k
L[i]>R[j]
A[k]=R[j]
++j,6++k
3 27 38 43 9 82
10
j
L R
i
3 9
0
A
1 2 3 4 5
k
L[i]>R[j]
A[k]=R[j]
++j,6++k
Merge Sort: Merge Operation
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k
L[i]<=R[j]
A[k]=L[i]
++i, ++k
3 27 38 43 9 10 82
j
L R
i
3 9 10
0
A
1 2 3 4 5 6
L[i]<=R[j]
A[k]=L[i]
++i, ++k
3 27 38 43 9 10 82
j
L R
i
0
A
1 2 3 4
5
3 9 6 10 27
k
3 27 38 43 9 10
k
R 82
j
L
0
A
1
i
2 3 4
5
3 9 6 10 27 38
L[i]<=R[j]
A[k]=L[i]
++i, ++k
Merge Sort: Merge Operation
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k
3 27 38 43 9 10 82
j
L R
i
3 9 10 27 38 43
0
A
1 2 3 4 5
L[i]>R[j]
A[k]=R[j]
++j, +6+k
k
3 27 38 43 9 10 82
R
i j
0 1 2 3 4
5
3 9 6 10 27 38 43 82
L[i]>R[j]
A[k]=R[j]
++j, ++k
All Elements Sorted
Merge Sort: Merge Operation
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L
A

Merge Sort
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def mergesort(alist):
print("Splitting", alist)
if len(alist)>1:
mid=len(alist)//2
lefthalf=alist[:mid]
righthalf=alist[mid:]
mergesort(lefthalf)
mergesort(righthalf)
i,j,k=0,0,0
while (i<len(lefthalf)) and (j<len(righthalf)):
if lefthalf[i]<righthalf[j]:
alist[k]=lefthalf[i]
i=i+1
else:
alist[k]=righthalf[j]
j=j+1
k=k+1

List – Merge Sort
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while i<len(lefthalf):
alist[k]=lefthalf[i]
i=i+1
k=k+1
while j<len(righthalf):
alist[k]=righthalf[j]
j=j+1
k=k+1
print("Merging", alist)
t=list(map(int,input("Enter the List elements separated by
space:").split(" ")))
print("List before sorting is",t)
mergesort(t)
print("List after sorting is",t)

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List – Merge Sort
Output:
Enter the List elements separated by space:6 9 2 4 2
List before sorting is ['6', '9', '2', '4', '2']
Splitting ['6', '9', '2', '4', '2']
Splitting ['6', '9']
Splitting ['6']
Splitting ['9']
Merging ['6', '9']
Splitting ['2', '4', '2']
Splitting ['2']
Splitting ['4', '2']
Splitting ['4']
Splitting ['2']
Merging ['2', '4']
Merging ['2', '2', '4']
Merging ['2', '2', '4', '6', '9']
List after sorting is ['2', '2', '4', '6', '9']

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• Searching is the process of finding a desired element in set of
items. The desired element is called "target".
• Any search is said to be successful or unsuccessful depending
upon whether the element that is being searched is found or not.
• For efficient data manipulation need fast searching.
• Static Search Structure
• Built once and searched many times
• Insertions occur before any retrieval
• Dynamic Search Structure
• Insertion, deletion and search operations can occur simultaneously
Searching: Introduction

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• Based on the type of search operation, these algorithms are
generally classified into two categories:
• Sequential Search: In this, the list or array is traversed
sequentially and every element is checked.
• For example: Linear Search.
• Interval Search: These algorithms are specifically designed for
searching in sorted data-structures. These type of searching
algorithms are much more efficient than Linear Search as they
repeatedly target the centre of the search structure and divide
the search space in half.
• For Example: Binary Search.
Searching: Algorithm

• This is the simplest method for searching.
• In this technique of searching, finding a particular value in a list,
that consists of checking every one of its elements, one at a time
and in sequence, until the desired one is found
• This method can be performed on a sorted or an unsorted list.
Sequential Search
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Algorithm
• Step 1: Start the search from the first element and Check the key
element with each element of list x.
• Step 2: If element is found, return the index position of the key.
• Step 3: If element is not found, print element is not present.
Sequential Search
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• Searching in case of unsorted list begins from the 0th element
and continues until the element or the end of the list is reached.
• Example: Find 12
35 42 12 5
12 Found!
Head
Sequential Search: Unsorted List

• Example: Find 13
35 42 12 5
Head
Sequential Search: Unsorted List
13 Not Found!

• Example: Find 13
5 12 35 42
Head
Sequential Search: Sorted List
13 Not Found!

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Sequential Search: Implementation
def linearSearch(L,ele):
for i in range(0,len(L)):
if(L[i]==ele):
return i+1
break
else:
return -1
l=input("Enter the List elements separated by
space:").split(" ")
e=input("Enter the element to search:"))
result=linearSearch(l,e)
if result>=0:
print(e,'is present in position ',result)
else:
print(e,'is not found in the list'+str(l))
Output:
Enter List elements separated by space:4 5 1 8 9
Enter the element to search:8
8 is present in position 4

• Binary search is an efficient algorithm for finding an item from a
sorted list of items.
divide and conquer
• This algorithm is also an example of
approach.
• The most essential thing of this method is it requires the list to be
in sorted order..
Binary Search
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Binary Search
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Algorithm
Step 1: Obtain the middle element
Step 2: If it equals searched value, return position and go to step 4
Step 3: Other wise, two cases possible.
Step 3.1: If search value < middle element, repeat the
procedure for sub list before the middle element.
Step 3.2: If search value > middle element, repeat the
procedure for sub list after the middle element.
Step 4: Stop the algorithm.

6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0
lo
w
1 2 3 4 5 6 7 8 9 10 11 12 13 14
high
Example: Binary search to Find(33) in the given list of elements.
• Middle=(low+high)/2
=(0+14)/2
=7
Binary Search:Example
Initial value
low=0
high=n-1

Example: Binary search for Find(33).
• Middle=(low+high)/2
=(0+14)/2
=7
6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
low high
Middle
Binary Search:Example

Example: Binary search for Find(33).
• Middle Element >Find
• Check Lower Half i.e. before middle element
• Change high=middle-1
1 1 2 3 4 5 5 6 7 8 9 9 9 9
6
3 4 5 3 3 1 3 4 2 4 3 5 6 7
0
low
1 2 3 4 5 6
high
7 8 9 10 11 12 13 14
Binary Search:Example

Example: Binary search for Find(33).
• Middle=(low+high)/2
=(0+6)/2
=3
1 1 2 3 4 5 5 6 7 8 9 9 9 9
3 4 5 3 3 1 3 4 2 4 3 5 6 7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
6
low middle high
Binary Search:Example

Example: Binary search for Find(33).
• Middle Element <Find
• Check First Half of top half i.e. after middle element
• Change low=middle+1
6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0 1 2 3 4
low
5 6
high
7 8 9 10 11 12 13 14
Binary Search:Example

Example: Binary search for Find(33).
• Middle=(low+high)/2
=(4+6)/2
=5
6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
lowmiddlehigh
Binary Search:Example

Example: Binary search for Find(33).
• Middle Element >Find
• Check Lower Half of bottom half i.e. before middle element
• Change high=middle-1
8
2
1 3 4 6
5 7 10
9 11 12 14
13
0
1 1 2 3 4 5 5 6 7 8 9 9 9 9
3 4 5 3 3 1 3 4 2 4 3 5 6 7
6
low
high
Binary Search:Example

Example: Binary search for Find(33).
• Middle=(low+high)/2
=(4+4)/2
=4
6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Binary Search:Example
low
high
middle

Example: Binary search for Find(33).
• Middle==Find
• The element 33 is found
6
1
3
1
4
2
5
3
3
4
3
5
1
5
3
6
4
7
2
8
4
9
3
9
5
9
6
9
7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Binary Search:Example
low
high
middle

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Binary Search: Implementation
def binarysearch(t,s):
low=0
high=len(t)-1
while high >= low:
middle=(low+high)//2
if t[middle] < s:
low=middle+1
elif t[m] >s:
high=middle-1
else:
return middle
return -1
t=input("Enter the List elements separated by space:").split("
")
s=input("Enter the element to search:")
result=binarysearch(t,s)
if result >= 0:
print(s," found at position",result+1)
else:
print(s," not found in the lsit"+str(t))
Output:
Enter the List elements
separated by space: 4 5 6 7 8
Enter the element to search:
8
8 is found at position 5