Iare ds lecture_notes_2

LECTURE NOTES
ON
DATA STRUCTURES
Year : 2017 - 2018
Course Code : ACS102
Regulations : R16
Semester : I B.Tech II Semester
Branch : CSE / IT / ECE / EEE
Prepared by
Ms. B Padmaja
Associate Professor
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
Dundigal, Hyderabad - 500 043

DATA STRUCTURES
II Semester: CSE / ECE / EEE / IT
Course Code Category Hours / Week Credits Maximum Marks
ACS002 Foundation
L T P C CIA SEE Total
3 1 - 4 30 70 100
Contact Classes: 45 Tutorial Classes: 15 Practical Classes: Nil Total Classes: 60
COURSE OBJECTIVES:
The course should enable the students to:
I. Learn the basic techniques of algorithm analysis.
II. Demonstrate several searching and sorting algorithms.
III. Implement linear and non-linear data structures.
IV. Demonstrate various tree and graph traversal algorithms.
V. Analyze and choose appropriate data structure to solve problems in real world.
UNIT - 1 INTRODUCTION TO DATA STRUCTURES, SEARCHING AND SORTING
Basic concepts: Introduction to data structures, classification of data structures, operations on
data structures, abstract data type, algorithms, different approaches to design an
algorithm, recursive algorithms; Searching techniques: Linear search, binary search and Fibonacci
search; Sorting techniques: Bubble sort, selection sort, insertion sort, quick sort, merge sort, and
comparison of sorting algorithms.
UNIT - 2 LINEAR DATA STRUCTURES
Stacks: Primitive operations, implementation of stacks using Arrays, applications of stacks
arithmetic expression conversion and evaluation; Queues: Primitive operations; Implementation
of queues using Array, applications of linear queue, circular queue and double ended queue
(DEQUE).
UNIT - 3 LINKED LISTS
Linked lists: Introduction, singly linked list, representation of a linked list in memory, operations on a
Single linked list; Applications of linked lists: Polynomial representation and sparse matrix
manipulation.
Types of linked lists: Circular linked lists, doubly linked lists; Linked list representation and
operations of Stack, linked list representation and operations of queue.
UNIT - 4 NON LINEAR DATA STRUCTURES
Trees : Basic concept, binary tree, binary tree representation, array and linked representations, binary
tree traversal, binary search tree, tree variants, application of trees; Graphs: Basic concept, graph
terminology, graph implementation, graph traversals, Application of graphs, Priority Queue.
UNIT - 5 BINARY TREES AND HASHING
Binary search trees: Binary search trees, properties and operations; Balanced search trees:
AVL trees; Introduction to M - Way search trees, B trees; Hashing and collision: Introduction,
hash tables, hash functions, collisions, applications of hashing.
LIST OF REFERENCE BOOKS:
1. Y Daniel Liang, “Introduction to Programming using Python”, Pearson.
2. Benjamin Baka, David Julian, “Python Data Structures and Algorithms”, Packt Publishers,2017.
3. Rance D. Necaise, “Data Structures and Algorithms using Python”, Wiley Student Edition.

4. Martin Jones, “Python for Complete Beginners”, 2015.
5. Zed A. Shaw, “Learn Python the Hard Way: a very simple introduction to the terrifyingly
beautiful world of computers and code”, 3e, Addison-Wesley, 2014.
6. Hemant Jain, “Problem Solving in Data Structures and Algorithms using Python: programming
interview guide”, 2016.
WEB REFERENCES:
1. https://docs.python.org/3/tutorial/datastructures.html
2. http://interactivepython.org/runestone/static/pythonds/index.html
3. http://www.tutorialspoint.com/data_structures_algorithms
4. http://www.geeksforgeeks.org/data-structures/
5. http://www.studytonight.com/data-structures/
6. http://www.coursera.org/specializations/data-structures-algorithms

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UNIT – I INTRODUCTION TO DATA STRUCTURES,
SEARCHING AND SORTING
Basic Concepts: Introduction to Data Structures:
A data structure is a way of storing data in a computer so that it can be used efficiently and it will allow
the most efficient algorithm to be used. The choice of the data structure begins from the choice of an
abstract data type (ADT). A well-designed data structure allows a variety of critical operations to be
performed, using as few resources, both execution time and memory space, as possible. Data structure
introduction refers to a scheme for organizing data, or in other words it is an arrangement of data in
computer's memory in such a way that it could make the data quickly available to the processor for
required calculations.
A data structure should be seen as a logical concept that must address two fundamental concerns.
1. First, how the data will be stored, and
2. Second, what operations will be performed on it.
As data structure is a scheme for data organization so the functional definition of a data structure should
be independent of its implementation. The functional definition of a data structure is known as ADT
(Abstract Data Type) which is independent of implementation. The way in which the data is organized
affects the performance of a program for different tasks. Computer programmers decide which data
structures to use based on the nature of the data and the processes that need to be performed on that
data. Some of the more commonly used data structures include lists, arrays, stacks, queues, heaps, trees,
and graphs.
Classification of Data Structures:
Data structures can be classified as
 Simple data structure
 Compound data structure
 Linear data structure
 Non linear data structure
[Fig 1.1 Classification of Data Structures]
Simple Data Structure:
Simple data structure can be constructed with the help of primitive data structure. A primitive data

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structure used to represent the standard data types of any one of the computer languages. Variables,
arrays, pointers, structures, unions, etc. are examples of primitive data structures.
Compound Data structure:
Compound data structure can be constructed with the help of any one of the primitive data structure and
it is having a specific functionality. It can be designed by user. It can be classified as
 Linear data structure
 Non-linear data structure
Linear Data Structure:
Linear data structures can be constructed as a continuous arrangement of data elements in the memory.
It can be constructed by using array data type. In the linear Data Structures the relationship of adjacency
is maintained between the data elements.
Operations applied on linear data structure:
The following list of operations applied on linear data structures
1. Add an element
2. Delete an element
3. Traverse
4. Sort the list of elements
5. Search for a data element
For example Stack, Queue, Tables, List, and Linked Lists.
Non-linear Data Structure:
Non-linear data structure can be constructed as a collection of randomly distributed set of data item
joined together by using a special pointer (tag). In non-linear Data structure the relationship of
adjacency is not maintained between the data items.
Operations applied on non-linear data structures:
The following list of operations applied on non-linear data structures.
1. Add elements
2. Delete elements
3. Display the elements
4. Sort the list of elements
5. Search for a data element
For example Tree, Decision tree, Graph and Forest
Abstract Data Type:
An abstract data type, sometimes abbreviated ADT, is a logical description of how we view the data
and the operations that are allowed without regard to how they will be implemented. This means that
we are concerned only with what data is representing and not with how it will eventually be
constructed. By providing this level of abstraction, we are creating an encapsulation around the data.
The idea is that by encapsulating the details of the implementation, we are hiding them from the user’s
view. This is called information hiding. The implementation of an abstract data type, often referred to
as a data structure, will require that we provide a physical view of the data using some collection of
programming constructs and primitive data types.

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[Fig. 1.2: Abstract Data Type (ADT)]
Algorithms:
Structure and Properties of Algorithm:
An algorithm has the following structure
1. Input Step
2. Assignment Step
3. Decision Step
4. Repetitive Step
5. Output Step
An algorithm is endowed with the following properties:
1. Finiteness: An algorithm must terminate after a finite number of steps.
2. Definiteness: The steps of the algorithm must be precisely defined or unambiguously specified.
3. Generality: An algorithm must be generic enough to solve all problems of a particular class.
4. Effectiveness: the operations of the algorithm must be basic enough to be put down on pencil and
paper. They should not be too complex to warrant writing another algorithm for the operation.
5. Input-Output: The algorithm must have certain initial and precise inputs, and outputs that may be
generated both at its intermediate and final steps.
Different Approaches to Design an Algorithm:
An algorithm does not enforce a language or mode for its expression but only demands adherence to its
properties.
Practical Algorithm Design Issues:
1. To save time (Time Complexity): A program that runs faster is a better program.
2. To save space (Space Complexity): A program that saves space over a competing program is

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considerable desirable.
Efficiency of Algorithms:
The performances of algorithms can be measured on the scales of time and space. The performance of
a program is the amount of computer memory and time needed to run a program. We use two
approaches to determine the performance of a program. One is analytical and the other is experimental.
In performance analysis we use analytical methods, while in performance measurement we conduct
experiments.
Time Complexity: The time complexity of an algorithm or a program is a function of the running time
of the algorithm or a program. In other words, it is the amount of computer time it needs to run to
completion.
Space Complexity: The space complexity of an algorithm or program is a function of the space needed
by the algorithm or program to run to completion.
The time complexity of an algorithm can be computed either by an empirical or theoretical approach.
The empirical or posteriori testing approach calls for implementing the complete algorithms and
executing them on a computer for various instances of the problem. The time taken by the execution of
the programs for various instances of the problem are noted and compared. The algorithm whose
implementation yields the least time is considered as the best among the candidate algorithmic
solutions.
Analyzing Algorithms
Suppose M is an algorithm, and suppose n is the size of the input data. Clearly the complexity f(n) of M
increases as n increases. It is usually the rate of increase of f(n) with some standard functions. The most
common computing times are
O(1), O(log2 n), O(n), O(n log2 n), O(n2
), O(n3
), O(2n
)
Example:

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The total frequency counts of the program segments A, B and C given by 1, (3n+1) and (3n2
+3n+1)
respectively are expressed as O(1), O(n) and O(n2
). These are referred to as the time complexities of the
program segments since they are indicative of the running times of the program segments. In a similar
manner space complexities of a program can also be expressed in terms of mathematical notations,

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which is nothing but the amount of memory they require for their execution.
Asymptotic Notations:
It is often used to describe how the size of the input data affects an algorithm’s usage of computational
resources. Running time of an algorithm is described as a function of input size n for large n.
Big oh(O): Definition: f(n) = O(g(n)) (read as f of n is big oh of g of n) if there exist a positive integer
n0 and a positive number c such that |f(n)| ≤ c|g(n)| for all n ≥ n0 . Here g(n) is the upper bound of the
function f(n).
Omega(Ω): Definition: f(n) = Ω(g(n)) ( read as f of n is omega of g of n), if there exists a positive
integer n0 and a positive number c such that |f(n)| ≥ c |g(n)| for all n ≥ n0. Here g(n) is the lower bound
of the function f(n).
Theta(Θ): Definition: f(n) = Θ(g(n)) (read as f of n is theta of g of n), if there exists a positive integer
n0 and two positive constants c1 and c2 such that c1 |g(n)| ≤ |f(n)| ≤ c2 |g(n)| for all n ≥ n0. The function
g(n) is both an upper bound and a lower bound for the function f(n) for all values of n, n ≥ n0 .

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Little oh(o): Definition: f(n) = O(g(n)) ( read as f of n is little oh of g of n), if f(n) = O(g(n)) and f(n) ≠
Ω(g(n)).
Time Complexity:
Time Complexities of various Algorithms:
Numerical Comparision of Different Algorithms:
S.No. log2n n nlog2n n2
n3
2n
1. 0 1 1 1 1 2
2. 1 2 2 4 8 4
3. 2 4 8 16 64 16
4. 3 8 24 64 512 256
5. 4 16 64 256 4096 65536
Reasons for analyzing algorithms:
1. To predict the resources that the algorithm requires

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 Computational Time(CPU consumption).
 Memory Space(RAM consumption).
 Communication bandwidth consumption.
2. To predict the running time of an algorithm
 Total number of primitive operations executed.
Recursive Algorithms:
GCD Design: Given two integers a and b, the greatest common divisor is recursively found using the
formula
gcd(a,b) = a if b=0
b if a=0
gcd(b, a mod b)
Fibonacci Design: To start a fibonacci series, we need to know the first two numbers.
Fibonacci(n) = 0 if n=0
1 if n=1
Fibonacci(n-1) + fibonacci(n-2)
Difference between Recursion and Iteration:
1. A function is said to be recursive if it calls itself again and again within its body whereas iterative
functions are loop based imperative functions.
2. Reursion uses stack whereas iteration does not use stack.
3. Recursion uses more memory than iteration as its concept is based on stacks.
4. Recursion is comparatively slower than iteration due to overhead condition of maintaining stacks.
5. Recursion makes code smaller and iteration makes code longer.
6. Iteration terminates when the loop-continuation condition fails whereas recursion terminates when
a base case is recognized.
7. While using recursion multiple activation records are created on stack for each call where as in
iteration everything is done in one activation record.
8. Infinite recursion can crash the system whereas infinite looping uses CPU cycles repeatedly.
9. Recursion uses selection structure whereas iteration uses repetetion structure.
Types of Recursion:
Recursion is of two types depending on whether a function calls itself from within itself or whether two
functions call one another mutually. The former is called direct recursion and the later is called
Base case
General case
Base case
General case

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indirect recursion. Thus there are two types of recursion:
 Direct Recursion
 Indirect Recursion
Recursion may be further categorized as:
 Linear Recursion
 Binary Recursion
 Multiple Recursion
Linear Recursion:
It is the most common type of Recursion in which function calls itself repeatedly until base condition
[termination case] is reached. Once the base case is reached the results are return to the caller function.
If a recursive function is called only once then it is called a linear recursion.
Binary Recursion:
Some recursive functions don't just have one call to themselves; they have two (or more). Functions
with two recursive calls are referred to as binary recursive functions.
Example1: The Fibonacci function fib provides a classic example of binary recursion. The Fibonacci
numbers can be defined by the rule:
fib(n) = 0 if n is 0,
= 1 if n is 1,
= fib(n-1) + fib(n-2) otherwise
For example, the first seven Fibonacci numbers are
Fib(0) = 0
Fib(1) = 1

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Fib(2) = Fib(1) + Fib(0) = 1
Fib(3) = Fib(2) + Fib(1) = 2
Fib(4) = Fib(3) + Fib(2) = 3
Fib(5) = Fib(4) + Fib(3) = 5
Fib(6) = Fib(5) + Fib(4) = 8
# Program to display the Fibonacci sequence up to n-th term where n is provided by the user
# change this value for a different result
nterms = 10
# uncomment to take input from the user
#nterms = int(input("How many terms? "))
# first two terms
n1 = 0
n2 = 1
count = 0
# check if the number of terms is valid
if nterms <= 0:
print("Please enter a positive integer")
elif nterms == 1:
print("Fibonacci sequence upto",nterms,":")
print(n1)
else:

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print("Fibonacci sequence upto",nterms,":")
while count < nterms:
print(n1,end=' , ')
nth = n1 + n2
# update values
n1 = n2
n2 = nth
count += 1
Tail Recursion:
Tail recursion is a form of linear recursion. In tail recursion, the recursive call is the last thing the
function does. Often, the value of the recursive call is returned. As such, tail recursive functions can
often be easily implemented in an iterative manner; by taking out the recursive call and replacing it with
a loop, the same effect can generally be achieved. In fact, a good compiler can recognize tail recursion
and convert it to iteration in order to optimize the performance of the code.
A good example of a tail recursive function is a function to compute the GCD, or Greatest Common
Denominator, of two numbers:
def factorial(n):
if n == 0: return 1
else: return factorial(n-1) * n
def tail_factorial(n, accumulator=1):
if n == 0: return 1
else: return tail_factorial(n-1, accumulator * n)
Recursive algorithms for Factorial, GCD, Fibonacci Series and Towers of Hanoi:
Factorial(n)
Input: integer n ≥ 0
Output: n!
1. If n = 0 then return (1)
2. else return prod(n, factorial(n − 1))
GCD(m, n)
Input: integers m > 0, n ≥ 0
Output: gcd (m, n)

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1. If n = 0 then return (m)
2. else return gcd(n,m mod n)
Time-Complexity: O(ln n)
Fibonacci(n)
Input: integer n ≥ 0
Output: Fibonacci Series: 1 1 2 3 5 8 13………………………………..
1. if n=1 or n=2
2. then Fibonacci(n)=1
3. else Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)
Towers of Hanoi
Input: The aim of the tower of Hanoi problem is to move the initial n different sized disks from needle
A to needle C using a temporary needle B. The rule is that no larger disk is to be placed above the
smaller disk in any of the needle while moving or at any time, and only the top of the disk is to be
moved at a time from any needle to any needle.
Output:
1. If n=1, move the single disk from A to C and return,
2. If n>1, move the top n-1 disks from A to B using C as temporary.
3. Move the remaining disk from A to C.
4. Move the n-1 disk disks from B to C, using A as temporary.
def TowerOfHanoi(n , from_rod, to_rod, aux_rod):

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if n == 1:
print "Move disk 1 from rod",from_rod,"to rod",to_rod
return
TowerOfHanoi(n-1, from_rod, aux_rod, to_rod)
print "Move disk",n,"from rod",from_rod,"to rod",to_rod
TowerOfHanoi(n-1, aux_rod, to_rod, from_rod)
n = 4
TowerOfHanoi(n, 'A', 'C', 'B')
Searching Techniques:
Linear Search: Searching is a process of finding a particular data item from a collection of data items
based on specific criteria. Every day we perform web searches to locate data items containing in various
pages. A search typically performed using a search key and it answers either True or False based on the
item is present or not in the list. Linear search algorithm is the most simplest algorithm to do sequential
search and this technique iterates over the sequence and checks one item at a time, until the desired item
is found or all items have been examined. In Python the in operator is used to find the desired item in a
sequence of items. The in operator makes searching task simpler and hides the inner working details.
Consider an unsorted single dimensional array of integers and we need to check whether 31 is present in
the array or not, then search begins with the first element. As the first element doesn't contain the desired
value, then the next element is compared to value 31 and this process continues until the desired element
is found in the sixth position. Similarly, if we want to search for 8 in the same array, then the search
begins in the same manner, starting with the first element until the desired element is found. In linear
search, we cannot determine that a given search value is present in the sequence or not until the entire
array is traversed.

14
Source Code:
def linear_search(obj, item, start=0):
for i in range(start, len(obj)):
if obj[i] == item:
return i
return -1
arr=[1,2,3,4,5,6,7,8]
x=4
result=linear_search(arr,x)
if result==-1:
print ("element does not exist")
else:
print ("element exist in position %d" %result)
Time Complexity of Linear Search:
Any algorithm is analyzed based on the unit of computation it performs. For linear search, we need to
count the number of comparisons performed, but each comparison may or may not search the desired
item.
Case Best Case Worst Case Average Case
If item is present 1 n n / 2
If item is not present n n n
Binary Search: In Binary search algorithm, the target key is examined in a sorted sequence and this
algorithm starts searching with the middle item of the sorted sequence.
a. If the middle item is the target value, then the search item is found and it returns True.
b. If the target item < middle item, then search for the target value in the first half of the list.
c. If the target item > middle item, then search for the target value in the second half of the list.
In binary search as the list is ordered, so we can eliminate half of the values in the list in each iteration.
Consider an example, suppose we want to search 10 in a sorted array of elements, then we first determine

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the middle element of the array. As the middle item contains 18, which is greater than the target value 10,
so can discard the second half of the list and repeat the process to first half of the array. This process is
repeated until the desired target item is located in the list. If the item is found then it returns True,
otherwise False.
Searching for 10 in a sorted array using Binary Search
Source Code:
array =[1,2,3,4,5,6,7,8,9]
def binary_search(searchfor,array):
lowerbound=0
upperbound=len(array)-1
found=False
while found==False and lowerbound<=upperbound:
midpoint=(lowerbound+upperbound)//2
if array[midpoint]==searchfor:
found =True
return found
elif array[midpoint]<searchfor:
lowerbound=midpoint+1
else:
upperbound=midpoint-1
return found
searchfor=int(input("what are you searching for?"))
if binary_search(searchfor,array):
print ("element found")
else:
print ("element not found")
Time Complexity of Binary Search:
In Binary Search, each comparison eliminates about half of the items from the list. Consider a list with n
items, then about n/2 items will be eliminated after first comparison. After second comparison, n/4 items

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of the list will be eliminated. If this process is repeated for several times, then there will be just one item
left in the list. The number of comparisons required to reach to this point is n/2i
= 1. If we solve for i, then
it gives us i = log n. The maximum number is comparison is logarithmic in nature, hence the time
complexity of binary search is O(log n).
Case Best Case Worst Case Average Case
If item is present 1 O(log n) O(log n)
If item is not present O(log n) O(log n) O(log n)
Fibonacci Search: It is a comparison based technique that uses Fibonacci numbers to search an element
in a sorted array. It follows divide and conquer approach and it has a O(log n) time complexity. Let the
element to be searched is x, then the idea is to first find the smallest Fibonacci number that is greater than
or equal to length of given array. Let the Fibonacci number be fib(nth
Fibonacci number). Use (n-2)th
Fibonacci number as index and say it is i, then compare a[i] with x, if x is same then return i. Else if x is
greater, then search the sub array after i, else search the sub array before i.
Source Code:
# Python3 program for Fibonacci search.
from bisect import bisect_left
# Returns index of x if present, else
# returns -1
def fibMonaccianSearch(arr, x, n):
# Initialize fibonacci numbers
fibMMm2 = 0 # (m-2)'th Fibonacci No.
fibMMm1 = 1 # (m-1)'th Fibonacci No.
fibM = fibMMm2 + fibMMm1 # m'th Fibonacci
# fibM is going to store the smallest
# Fibonacci Number greater than or equal to n
while (fibM < n):
fibMMm2 = fibMMm1
fibMMm1 = fibM
fibM = fibMMm2 + fibMMm1
# Marks the eliminated range from front
offset = -1;
# while there are elements to be inspected.
# Note that we compare arr[fibMm2] with x.
# When fibM becomes 1, fibMm2 becomes 0
while (fibM > 1):

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# Check if fibMm2 is a valid location
i = min(offset+fibMMm2, n-1)
# If x is greater than the value at
# index fibMm2, cut the subarray array
# from offset to i
if (arr[i] < x):
fibM = fibMMm1
fibMMm1 = fibMMm2
fibMMm2 = fibM - fibMMm1
offset = i
# If x is greater than the value at
# index fibMm2, cut the subarray
# after i+1
elif (arr[i] > x):
fibM = fibMMm2
fibMMm1 = fibMMm1 - fibMMm2
fibMMm2 = fibM - fibMMm1
# element found. return index
else :
return i
# comparing the last element with x */
if(fibMMm1 and arr[offset+1] == x):
return offset+1;
# element not found. return -1
return -1
# Driver Code
arr = [10, 22, 35, 40, 45, 50, 80, 82, 85, 90, 100]
n = len(arr)
x = 80
print("Found at index:",
fibMonaccianSearch(arr, x, n))
Time Complexity of Fibonacci Search:
Time complexity for Fibonacci search is O(log2 n)
Sorting Techniques:
Sorting in general refers to various methods of arranging or ordering things based on criteria's
(numerical, chronological, alphabetical, hierarchical etc.). There are many approaches to sorting data and
each has its own merits and demerits.

18
Bubble Sort:
This sorting technique is also known as exchange sort, which arranges values by iterating over the list
several times and in each iteration the larger value gets bubble up to the end of the list. This algorithm
uses multiple passes and in each pass the first and second data items are compared. if the first data item is
bigger than the second, then the two items are swapped. Next the items in second and third position are
compared and if the first one is larger than the second, then they are swapped, otherwise no change in
their order. This process continues for each successive pair of data items until all items are sorted.
Bubble Sort Algorithm:
Step 1: Repeat Steps 2 and 3 for i=1 to 10
Step 2: Set j=1
Step 3: Repeat while j<=n
(A) if a[i] < a[j]
Then interchange a[i] and a[j]
[End of if]
(B) Set j = j+1
[End of Inner Loop]
[End of Step 1 Outer Loop]
Step 4: Exit

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Various Passes of Bubble Sort
Source Code:
# Python program for implementation of Bubble Sort
def bubbleSort(arr):
n = len(arr)
# Traverse through all array elements
for i in range(n):
# Last i elements are already in place
for j in range(0, n-i-1):
# traverse the array from 0 to n-i-1
# Swap if the element found is greater
# than the next element
if arr[j] > arr[j+1] :

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arr[j], arr[j+1] = arr[j+1], arr[j]
# Driver code to test above
arr = [64, 34, 25, 12, 22, 11, 90]
bubbleSort(arr)
print ("Sorted array is:")
for i in range(len(arr)):
print ("%d" %arr[i])
Step-by-step example:
Let us take the array of numbers "5 1 4 2 8", and sort the array from lowest number to greatest number
using bubble sort. In each step, elements written in bold are being compared. Three passes will be
required.
First Pass:
( 5 1 4 2 8 ) ( 1 5 4 2 8 ), Here, algorithm compares the first two elements, and swaps since 5 > 1.
( 1 5 4 2 8 ) ( 1 4 5 2 8 ), Swap since 5 > 4
( 1 4 5 2 8 ) ( 1 4 2 5 8 ), Swap since 5 > 2
( 1 4 2 5 8 )
( 1 4 2 5 8 ), Now, since these elements are already in order (8 > 5), algorithm does not swap them.
Second Pass:
( 1 4 2 5 8 ) ( 1 4 2 5 8 )
( 1 4 2 5 8 ) ( 1 2 4 5 8 ), Swap since 4 > 2
( 1 2 4 5 8 ) ( 1 2 4 5 8 )
( 1 2 4 5 8 ) ( 1 2 4 5 8 )
Now, the array is already sorted, but our algorithm does not know if it is completed. The algorithm needs
one whole pass without any swap to know it is sorted.
Third Pass:
( 1 2 4 5 8 ) ( 1 2 4 5 8 )
( 1 2 4 5 8 ) ( 1 2 4 5 8 )
( 1 2 4 5 8 ) ( 1 2 4 5 8 )

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( 1 2 4 5 8 ) ( 1 2 4 5 8 )
Time Complexity:
The efficiency of Bubble sort algorithm is independent of number of data items in the array and its initial
arrangement. If an array containing n data items, then the outer loop executes n-1 times as the algorithm
requires n-1 passes. In the first pass, the inner loop is executed n-1 times; in the second pass, n-2 times; in
the third pass, n-3 times and so on. The total number of iterations resulting in a run time of O(n2
).
Worst Case Performance O(n2
)
Best Case Performance O(n2
)
Average Case Performance O(n2
)
Selection Sort:
Selection sort algorithm is one of the simplest sorting algorithm, which sorts the elements in an array by
finding the minimum element in each pass from unsorted part and keeps it in the beginning. This
sorting technique improves over bubble sort by making only one exchange in each pass. This sorting
technique maintains two sub arrays, one sub array which is already sorted and the other one which is
unsorted. In each iteration the minimum element (ascending order) is picked from unsorted array and
moved to sorted sub array..
Selection Sort Algorithm:
Source Code:
# Python program for implementation of Selection
# Sort
import sys
A = [64, 25, 12, 22, 11]
# Traverse through all array elements
for i in range(len(A)):

22
# Find the minimum element in remaining
# unsorted array
min_idx = i
for j in range(i+1, len(A)):
if A[min_idx] > A[j]:
min_idx = j
# Swap the found minimum element with
# the first element
A[i], A[min_idx] = A[min_idx], A[i]
print ("Sorted array")
for i in range(len(A)):
print("%d" %A[i])
Output:
Enter array size:6
Enter the elements:96 94 81 56 76 45
The elements after sorting are: 45 56 76 81 94 96
Here is an example of this sort algorithm sorting five elements:
64 25 12 22 11
11 25 12 22 64
11 12 25 22 64
11 12 22 25 64
11 12 22 25 64
Time Complexity:
Selection sort is not difficult to analyze compared to other sorting algorithms since none of the loops
depend on the data in the array. Selecting the lowest element requires scanning all n elements (this takes
n − 1 comparisons) and then swapping it into the first position. Finding the next lowest element requires
scanning the remaining n − 1 elements and so on, for (n − 1) + (n − 2) + ... + 2 + 1 = n(n − 1) / 2 ∈ O(n2
)
comparisons. Each of these scans requires one swap for n − 1 elements (the final element is already in
place).
)
Best Case Performance O(n2
)

23
)
Insertion Sort:
An algorithm consider the elements one at a time, inserting each in its suitable place among those already
considered (keeping them sorted). Insertion sort is an example of an incremental algorithm. It builds the
sorted sequence one number at a time. This is a suitable sorting technique in playing card games.
Insertion sort provides several advantages:
 Simple implementation
 Efficient for (quite) small data sets
 Adaptive (i.e., efficient) for data sets that are already substantially sorted: the time complexity is
O(n + d), where d is the number of inversions
 More efficient in practice than most other simple quadratic (i.e., O(n2
)) algorithms such
as selection sort or bubble sort; the best case (nearly sorted input) is O(n)
 Stable; i.e., does not change the relative order of elements with equal keys
 In-place; i.e., only requires a constant amount O(1) of additional memory space
 Online; i.e., can sort a list as it receives it
Source Code:
# Python program for implementation of Insertion Sort
# Function to do insertion sort
def insertionSort(arr):
# Traverse through 1 to len(arr)
for i in range(1, len(arr)):

24
key = arr[i]
# Move elements of arr[0..i-1], that are
# greater than key, to one position ahead
# of their current position
j = i-1
while j >=0 and key < arr[j] :
arr[j+1] = arr[j]
j -= 1
arr[j+1] = key
arr = [12, 11, 13, 5, 6]
insertionSort(arr)
for i in range(len(arr)):

25
Suppose, you want to sort elements in ascending as in above figure. Then,
1. The second element of an array is compared with the elements that appear before it (only first
element in this case). If the second element is smaller than first element, second element is
inserted in the position of first element. After first step, first two elements of an array will be
sorted.
2. The third element of an array is compared with the elements that appears before it (first and
second element). If third element is smaller than first element, it is inserted in the position of first
element. If third element is larger than first element but, smaller than second element, it is
inserted in the position of second element. If third element is larger than both the elements, it is
kept in the position as it is. After second step, first three elements of an array will be sorted.
3. Similarly, the fourth element of an array is compared with the elements that appear before it (first,
second and third element) and the same procedure is applied and that element is inserted in the
proper position. After third step, first four elements of an array will be sorted.
If there are n elements to be sorted. Then, this procedure is repeated n-1 times to get sorted list of array.
Time Complexity:
)
Best Case Performance(nearly) O(n)
)
Output:
Enter no of elements:5
Enter elements:1 65 0 32 66
Elements after sorting: 0 1 32 65 66
Quick Sort :
Quick sort is a divide and conquer algorithm. Quick sort first divides a large list into two smaller sub-
lists: the low elements and the high elements. Quick sort can then recursively sort the sub-lists.
The steps are:
1. Pick an element, called a pivot, from the list.
2. Reorder the list so that all elements with values less than the pivot come before the pivot, while
all elements with values greater than the pivot come after it (equal values can go either way).
After this partitioning, the pivot is in its final position. This is called the partition operation.
3. Recursively apply the above steps to the sub-list of elements with smaller values and separately
the sub-list of elements with greater values.
The base case of the recursion is lists of size zero or one, which never need to be sorted.

26
Quick sort, or partition-exchange sort, is a sorting algorithm developed by Tony Hoare that, on
average, makes O(n log n) comparisons to sort n items. In the worst case, it makes O(n2
) comparisons,
though this behavior is rare. Quick sort is often faster in practice than other O(n log n) algorithms. It
works by first of all by partitioning the array around a pivot value and then dealing with the 2 smaller
partitions separately. Partitioning is the most complex part of quick sort. The simplest thing is to use the
first value in the array, a[l] (or a[0] as l = 0 to begin with) as the pivot. After the partitioning, all values to
the left of the pivot are <= pivot and all values to the right are > pivot. The same procedure for the two
remaining sub lists is repeated and so on recursively until we have the entire list sorted.
Advantages:
 One of the fastest algorithms on average.
 Does not need additional memory (the sorting takes place in the array - this is called in-place
processing).
Disadvantages: The worst-case complexity is O(N2
)
Source Code:
# Python program for implementation of Quicksort Sort
# This function takes last element as pivot, places
# the pivot element at its correct position in sorted
# array, and places all smaller (smaller than pivot)
# to left of pivot and all greater elements to right
# of pivot
def partition(arr,low,high):
i = ( low-1 ) # index of smaller element
pivot = arr[high] # pivot
for j in range(low , high):
# If current element is smaller than or
# equal to pivot
if arr[j] <= pivot:
# increment index of smaller element
i = i+1
arr[i],arr[j] = arr[j],arr[i]
arr[i+1],arr[high] = arr[high],arr[i+1]
return ( i+1 )
# The main function that implements QuickSort
# arr[] --> Array to be sorted,
# low --> Starting index,
# high --> Ending index

27
# Function to do Quick sort
def quickSort(arr,low,high):
if low < high:
# pi is partitioning index, arr[p] is now
# at right place
pi = partition(arr,low,high)
# Separately sort elements before
# partition and after partition
quickSort(arr, low, pi-1)
quickSort(arr, pi+1, high)
arr = [10, 7, 8, 9, 1, 5]
n = len(arr)
quickSort(arr,0,n-1)
for i in range(n):
1 2 3 4 5 6 7 8 9 10 11 12 13 Remarks
38 08 16 06 79 57 24 56 02 58 04 70 45
Pivot 08 16 06 Up 57 24 56 02 58 Dn 70 45 Swap up and
down
Pivot 08 16 06 04 57 24 56 02 58 79 70 45
Pivot 08 16 06 04 Up 24 56 Dn 58 79 70 45 Swap up and
down
Pivot 08 16 06 04 02 24 56 57 58 79 70 45
Pivot 08 16 06 04 02 Dn Up 57 58 79 70 45 Swap pivot
and down
24 08 16 06 04 02 38 56 57 58 79 70 45
Pivot 08 16 06 04 Dn Up 56 57 58 79 70 45 Swap pivot
and down
(02 08 16 06 04 24) 38 (56 57 58 79 70 45)
Pivot 08 16 06 04 Up

28
dn
Pivot Up 06 Dn Swap up and
down
Pivot 04 06 16
Pivot 04 Dn Up Swap pivot
and down
06 04 08
Pivot Dn Up Swap pivot
and down
04 06
(02 04 06 08 16 24 38) (56 57 58 79 70 45)
Pivot Up 58 79 70 Dn Swap up and
down
Pivot 45 58 79 70 57
Pivot Dn Up 79 70 57 Swap pivot
and down
(45) 56 (58 79 70 57)
Pivot Up 70 Dn Swap up and
down
Pivot 57 70 79
Pivot Dn Up 79 Swap down
and pivot
(57) 58 (70 79)
Pivot
Dn
Up Swap pivot
and down
02 04 06 08 16 24 38 45 56 57 58 70 79 The array is
sorted
Time Complexity:
)
Best Case Performance(nearly) O(n log2 n)
Average Case Performance O(n log2 n)

29
Merge Sort:
Merge sort is based on Divide and conquer method. It takes the list to be sorted and divide it in half to
create two unsorted lists. The two unsorted lists are then sorted and merged to get a sorted list. The two
unsorted lists are sorted by continually calling the merge-sort algorithm; we eventually get a list of size
1 which is already sorted. The two lists of size 1 are then merged.
Merge Sort Procedure:
This is a divide and conquer algorithm.
This works as follows :
1. Divide the input which we have to sort into two parts in the middle. Call it the left part and right part.
2. Sort each of them separately. Note that here sort does not mean to sort it using some other method.
We use the same function recursively.
3. Then merge the two sorted parts.
Input the total number of elements that are there in an array (number_of_elements). Input the array
(array[number_of_elements]). Then call the function MergeSort() to sort the input array. MergeSort()
function sorts the array in the range [left,right] i.e. from index left to index right inclusive. Merge()
function merges the two sorted parts. Sorted parts will be from [left, mid] and [mid+1, right]. After
merging output the sorted array.
MergeSort() function:
It takes the array, left-most and right-most index of the array to be sorted as arguments. Middle index
(mid) of the array is calculated as (left + right)/2. Check if (left<right) cause we have to sort only when
left<right because when left=right it is anyhow sorted. Sort the left part by calling MergeSort() function
again over the left part MergeSort(array,left,mid) and the right part by recursive call of MergeSort
function as MergeSort(array,mid + 1, right). Lastly merge the two arrays using the Merge function.
Merge() function:
It takes the array, left-most , middle and right-most index of the array to be merged as arguments.
Finally copy back the sorted array to the original array.
Source Code:
# Recursive Python Program for merge sort
def merge(left, right):
if not len(left) or not len(right):
return left or right
result = []
i, j = 0, 0
while (len(result) < len(left) + len(right)):
if left[i] < right[j]:
result.append(left[i])

30
i+= 1
else:
result.append(right[j])
j+= 1
if i == len(left) or j == len(right):
result.extend(left[i:] or right[j:])
break
return result
def mergesort(list):
if len(list) < 2:
return list
middle = int(len(list)/2)
left = mergesort(list[:middle])
right = mergesort(list[middle:])
return merge(left, right)
seq = [12, 11, 13, 5, 6, 7]
print("Given array is")
print(seq);
print("n")
print("Sorted array is")
print(mergesort(seq))

31
Merge Sort Example
Time Complexity:
Worst Case Performance O(n log2 n)
Best Case Performance(nearly) O(n log2 n)
Average Case Performance O(n log2 n)
Comparison of Sorting Algorithms:

32
Time Complexity comparison of Sorting Algorithms:
Algorithm Data Structure Time Complexity
Best Average Worst
Quicksort Array O(n log(n)) O(n log(n)) O(n^2)
Mergesort Array O(n log(n)) O(n log(n)) O(n log(n))
Bubble Sort Array O(n) O(n^2) O(n^2)
Insertion Sort Array O(n) O(n^2) O(n^2)
Select Sort Array O(n^2) O(n^2) O(n^2)
Space Complexity comparison of Sorting Algorithms:
Algorithm Data Structure
Worst Case Auxiliary Space
Complexity
Quicksort Array O(n)
Mergesort Array O(n)
Bubble Sort Array O(1)
Insertion Sort Array O(1)
Select Sort Array O(1)
Bucket Sort Array O(nk)

33
UNIT – II LINEAR DATA STRUCTURES
Stacks Primitive Operations:
A stack is a container of objects that are inserted and removed according to the last-in first-out (LIFO)
principle. In the pushdown stacks only two operations are allowed: push the item into the stack, and pop
the item out of the stack. A stack is a limited access data structure - elements can be added and removed
from the stack only at the top. Push adds an item to the top of the stack, pop removes the item from the
top. A helpful analogy is to think of a stack of books; you can remove only the top book, also you can
add a new book on the top.
A stack may be implemented to have a bounded capacity. If the stack is full and does not contain enough
space to accept an entity to be pushed, the stack is then considered to be in an overflow state. The pop
operation removes an item from the top of the stack. A pop either reveals previously concealed items or
results in an empty stack, but, if the stack is empty, it goes into underflow state, which means no items
are present in stack to be removed.
Stack (ADT) Data Structure:
Stack is an Abstract data structure (ADT) works on the principle Last In First Out (LIFO). The last
element add to the stack is the first element to be delete. Insertion and deletion can be takes place at one
end called TOP. It looks like one side closed tube.
 The add operation of the stack is called push operation
 The delete operation is called as pop operation.
 Push operation on a full stack causes stack overflow.
 Pop operation on an empty stack causes stack underflow.
 SP is a pointer, which is used to access the top element of the stack.
 If you push elements that are added at the top of the stack;
 In the same way when we pop the elements, the element at the top of the stack is deleted.

34
Operations of stack:
There are two operations applied on stack they are
1. push
2. pop.
While performing push & pop operations the following test must be conducted on the stack.
1) Stack is empty or not
2) Stack is full or not
Push:
Push operation is used to add new elements in to the stack. At the time of addition first check the stack is
full or not. If the stack is full it generates an error message "stack overflow".
Pop:
Pop operation is used to delete elements from the stack. At the time of deletion first check the stack is
empty or not. If the stack is empty it generates an error message "stack underflow".
Representation of a Stack using Arrays:
Let us consider a stack with 6 elements capacity. This is called as the size of the stack. The number of
elements to be added should not exceed the maximum size of the stack. If we attempt to add new
element beyond the maximum size, we will encounter a stack overflow condition. Similarly, you cannot
remove elements beyond the base of the stack. If such is the case, we will reach a stack underflow
condition.
When a element is added to a stack, the operation is performed by push().
When an element is taken off from the stack, the operation is performed by pop().

35
Source code for stack operations, using array:
STACK: Stack is a linear data structure which works under the principle of last in first out. Basic
operations: push, pop, display.
1. PUSH: if (top==MAX), display Stack overflow else reading the data and making stack [top]
=data and incrementing the top value by doing top++.
2. POP: if (top==0), display Stack underflow else printing the element at the top of the stack and
decrementing the top value by doing the top.
3. DISPLAY: IF (TOP==0), display Stack is empty else printing the elements in the stack from
stack [0] to stack [top].
# Python program for implementation of stack
# import maxsize from sys module
# Used to return -infinite when stack is empty from sys import maxsize
# Function to create a stack. It initializes size of stack as 0
def createStack():
stack = []
return stack
# Stack is empty when stack size is 0
def isEmpty(stack):
return len(stack) == 0
# Function to add an item to stack. It increases size by 1
def push(stack, item):
stack.append(item)
print("pushed to stack " + item)
# Function to remove an item from stack. It decreases size by 1
def pop(stack):
if (isEmpty(stack)):
print("stack empty")

36
return str(-maxsize -1) #return minus infinite
return stack.pop()
# Driver program to test above functions
stack = createStack()
print("maximum size of array is",maxsize)
push(stack, str(10))
print(pop(stack) + " popped from stack")
Linked List Implementation of Stack:
We can represent a stack as a linked list. In a stack push and pop operations are performed at one end
called top. We can perform similar operations at one end of list using top pointer.
Source code for stack operations, using linked list:
# Python program for linked list implementation of stack
# Class to represent a node
class StackNode:
# Constructor to initialize a node
def __init__(self, data):
self.data = data

37
self.next = None
class Stack:
# Constructor to initialize the root of linked list
def __init__(self):
self.root = None
def isEmpty(self):
return True if self.root is None else False
def push(self, data):
newNode = StackNode(data)
newNode.next = self.root
self.root = newNode
print ("%d pushed to stack" %(data))
def pop(self):
if (self.isEmpty()):
return float("-inf")
temp = self.root
self.root = self.root.next
popped = temp.data
return popped
def peek(self):
if self.isEmpty():
return float("-inf")
return self.root.data
# Driver program to test above class
stack = Stack()
stack.push(10)
stack.push(20)
stack.push(30)
print ("%d popped from stack" %(stack.pop()))
print ("Top element is %d " %(stack.peek()))
Stack Applications:
1. Stack is used by compilers to check for balancing of parentheses, brackets and braces.
2. Stack is used to evaluate a postfix expression.
3. Stack is used to convert an infix expression into postfix/prefix form.

38
4. In recursion, all intermediate arguments and return values are stored on the processor‟s stack.
5. During a function call the return address and arguments are pushed onto a stack and on return
they are popped off.
6. Depth first search uses a stack data structure to find an element from a graph.
In-fix- to Postfix Transformation:
Procedure:
Procedure to convert from infix expression to postfix expression is as follows:
1. Scan the infix expression from left to right.
2. a) If the scanned symbol is left parenthesis, push it onto the stack.
b) If the scanned symbol is an operand, then place directly in the postfix expression
(output).
c) If the symbol scanned is a right parenthesis, then go on popping all the items from
the stack and place them in the postfix expression till we get the matching left
parenthesis.
d) If the scanned symbol is an operator, then go on removing all the operators from the
stack and place them in the postfix expression, if and only if the precedence of the
operator which is on the top of the stack is greater than (or equal) to the precedence
of the scanned operator and push the scanned operator onto the stack otherwise,
push the scanned operator onto the stack.
Convert the following infix expression A + B * C – D / E * H into its equivalent postfix expression.
Symbol Postfix string Stack Remarks
A A
+ A +
B A B +
* A B + *
C A B C -
- A B C * + -
D A B C * + D -
/ A B C * + D - /
E A B C * + D E - /
* A B C * + D E / - *

39
H A B C * + D E / H - *
End of
string A B C * + D E / H * -
The input is now empty. Pop the output symbols from the
stack until it is empty.
Source Code:
# Python program to convert infix expression to postfix
# Class to convert the expression
import string
class Conversion:
# Constructor to initialize the class variables
def __init__(self, capacity):
self.top = -1
self.capacity = capacity
# This array is used a stack
self.array = []
# Precedence setting
self.output = []
self.precedence = {'+':1, '-':1, '*':2, '/':2, '^':3}
# check if the stack is empty
def isEmpty(self):
return True if self.top == -1 else False
# Return the value of the top of the stack
def peek(self):
return self.array[-1]
# Pop the element from the stack
def pop(self):
if not self.isEmpty():
self.top -= 1
return self.array.pop()
else:
return "$"
# Push the element to the stack
def push(self, op):
self.top += 1
self.array.append(op)
# A utility function to check is the given character
# is operand
def isOperand(self, ch):
return ch.isalpha()
# Check if the precedence of operator is strictly
# less than top of stack or not

40
def notGreater(self, i):
try:
a = self.precedence[i]
b = self.precedence[self.peek()]
return True if a <= b else False
except KeyError:
return False
# The main function that converts given infix expression
# to postfix expression
def infixToPostfix(self, exp):
# Iterate over the expression for conversion
for i in exp:
# If the character is an operand,
# add it to output
if self.isOperand(i):
self.output.append(i)
# If the character is an '(', push it to stack
elif i == '(':
self.push(i)
# If the scanned character is an ')', pop and
# output from the stack until and '(' is found
elif i == ')':
while( (not self.isEmpty()) and self.peek() != '('):
a = self.pop()
self.output.append(a)
if (not self.isEmpty() and self.peek() != '('):
return -1
else:
self.pop()
# An operator is encountered
else:
while(not self.isEmpty() and self.notGreater(i)):
self.output.append(self.pop())
self.push(i)
# pop all the operator from the stack
while not self.isEmpty():
self.output.append(self.pop())
result= "".join(self.output)
print(result)
# Driver program to test above function
exp = "a+b*(c^d-e)^(f+g*h)-i"
obj = Conversion(len(exp))
obj.infixToPostfix(exp)

41
Evaluating Arithmetic Expressions:
Procedure:
The postfix expression is evaluated easily by the use of a stack. When a number is seen, it is pushed onto
the stack; when an operator is seen, the operator is applied to the two numbers that are popped from the
stack and the result is pushed onto the stack.
Evaluate the postfix expression: 6 5 2 3 + 8 * + 3 + *
Symbol Operand 1 Operand 2 Value Stack Remarks
6 6
5 6, 5
2 6, 5, 2
3 6, 5, 2, 3
The first four symbols are
placed on the stack.
+ 2 3 5 6, 5, 5
Next a „+‟ is read, so 3 and 2
are popped from the stack and
their sum 5, is pushed
8 2 3 5 6, 5, 5, 8 Next 8 is pushed
* 5 8 40 6, 5, 40
Now a „*‟ is seen, so 8 and 5
are popped as 8 * 5 = 40 is
pushed
+ 5 40 45 6, 45
Next, a „+‟ is seen, so 40 and
5 are popped and 40 + 5 = 45
is pushed
3 5 40 45 6, 45, 3 Now, 3 is pushed
+ 45 3 48 6, 48
Next, „+‟ pops 3 and 45 and
pushes 45 + 3 = 48 is pushed
* 6 48 288 288
Finally, a „*‟ is seen and 48
and 6 are popped, the result 6
* 48 = 288 is pushed
Source Code:
# Python program to evaluate value of a postfix expression
# Class to convert the expression
class Evaluate:
# Constructor to initialize the class variables
def __init__(self, capacity):
self.top = -1

42
self.capacity = capacity
# This array is used a stack
self.array = []
# check if the stack is empty
def isEmpty(self):
return True if self.top == -1 else False
# Return the value of the top of the stack
def peek(self):
return self.array[-1]
# Pop the element from the stack
def pop(self):
if not self.isEmpty():
self.top -= 1
return self.array.pop()
else:
return "$"
# Push the element to the stack
def push(self, op):
self.top += 1
self.array.append(op)
# The main function that converts given infix expression
# to postfix expression
def evaluatePostfix(self, exp):
# Iterate over the expression for conversion
for i in exp:
# If the scanned character is an operand
# (number here) push it to the stack
if i.isdigit():
self.push(i)
# If the scanned character is an operator,
# pop two elements from stack and apply it.
else:
val1 = self.pop()
val2 = self.pop()
self.push(str(eval(val2 + i + val1)))
return int(self.pop())
# Driver program to test above function
exp = "231*+9-"
obj = Evaluate(len(exp))
print ("Value of {0} is {1}".format(exp, obj.evaluatePostfix(exp)))

43
Basic Queue Operations:
A queue is a data structure that is best described as "first in, first out". A queue is another special kind of
list, where items are inserted at one end called the rear and deleted at the other end called the front. A
real world example of a queue is people waiting in line at the bank. As each person enters the bank, he
or she is "enqueued" at the back of the line. When a teller becomes available, they are "dequeued" at the
front of the line.
Representation of a Queue using Array:
Let us consider a queue, which can hold maximum of five elements. Initially the queue is empty.
F R
Que ue E mpty
F RO NT = REA R = 0
0 1 2 3 4
Now, insert 11 to the queue. Then queue status will be:
11
F R
REA R = REA R + 1 = 1
F RO NT = 0
0 1 2 3 4
Next, insert 22 to the queue. Then the queue status is:
11 22
F R
REA R = REA R + 1 = 2
F RO NT = 0
0 1 2 3 4
Again insert another element 33 to the queue. The status of the queue is:

44
11 22 33
F R
REA R = REA R + 1 = 3
F RO NT = 0
0 1 2 3 4
Now, delete an element. The element deleted is the element at the front of the queue. So the status of the
queue is:
22 33
F R
REA R = 3
F RO NT = F RO NT + 1 = 1
0 1 2 3 4
Again, delete an element. The element to be deleted is always pointed to by the FRONT pointer. So, 22
is deleted. The queue status is as follows:
33
F R
REA R = 3
F RO NT = F RO NT + 1 = 2
0 1 2 3 4
Now, insert new elements 44 and 55 into the queue. The queue status is:
33 44 55
F R
REA R = 5
F RO NT = 2
0 1 2 3 4
Next insert another element, say 66 to the queue. We cannot insert 66 to the queue as the rear crossed the
maximum size of the queue (i.e., 5). There will be queue full signal. The queue status is as follows:

45
33 44 55
F R
REA R = 5
F RO NT = 2
0 1 2 3 4
Now it is not possible to insert an element 66 even though there are two vacant positions in the linear
queue. To over come this problem the elements of the queue are to be shifted towards the beginning of
the queue so that it creates vacant position at the rear end. Then the FRONT and REAR are to be
adjusted properly. The element 66 can be inserted at the rear end. After this operation, the queue status is
as follows:
33 44 55 66
F R
REA R = 4
F RO NT = 0
0 1 2 3 4
This difficulty can overcome if we treat queue position with index 0 as a position that comes after
position with index 4 i.e., we treat the queue as a circular queue.
Procedure for Queue operations using array:
In order to create a queue we require a one dimensional array Q(1:n) and two variables front and rear.
The conventions we shall adopt for these two variables are that front is always 1 less than the actual
front of the queue and rear always points to the last element in the queue. Thus, front = rear if and only if
there are no elements in the queue. The initial condition then is front = rear = 0.
The various queue operations to perform creation, deletion and display the elements in a queue are as
follows:
1. insertQ(): inserts an element at the end of queue Q.
2. deleteQ(): deletes the first element of Q.
3. displayQ(): displays the elements in the queue.
Linked List Implementation of Queue: We can represent a queue as a linked list. In a queue data is
deleted from the front end and inserted at the rear end. We can perform similar operations on the two
ends of a list. We use two pointers front and rear for our linked queue implementation.

46
Source Code:
front = 0
rear = 0
mymax = 3
# Function to create a stack. It initializes size of stack as 0
def createQueue():
queue = []
return queue
# Stack is empty when stack size is 0
def isEmpty(queue):
return len(queue) == 0
# Function to add an item to stack. It increases size by 1
def enqueue(queue,item):
queue.append(item)
# Function to remove an item from stack. It decreases size by 1
def dequeue(queue):
if (isEmpty(queue)):
return "Queue is empty"
item=queue[0]
del queue[0]
return item
# Driver program to test above functions
queue = createQueue()
while True:
print("1 Enqueue")
print("2 Dequeue")
print("3 Display")
print("4 Quit")

47
ch=int(input("Enter choice"))
if(ch==1):
if(rear < mymax):
item=input("enter item")
enqueue(queue, item)
rear = rear + 1
else:
print("Queue is full")
elif(ch==2):
print(dequeue(queue))
elif(ch==3):
print(queue)
else:
break
Applications of Queues:
1. It is used to schedule the jobs to be processed by the CPU.
2. When multiple users send print jobs to a printer, each printing job is kept in the printing queue.
Then the printer prints those jobs according to first in first out (FIFO) basis.
3. Breadth first search uses a queue data structure to find an element from a graph.
Disadvantages of Linear Queue:
There are two problems associated with linear queue. They are:
 Time consuming: linear time to be spent in shifting the elements to the beginning of the queue.
 Signaling queue full: even if the queue is having vacant position.
Round Robin Algorithm:
The round-robin (RR) scheduling algorithm is designed especially for time-sharing systems. It is similar
to FCFS scheduling, but pre-emption is added to switch between processes. A small unit of time, called
a time quantum or time slices, is defined. A time quantum is generally from 10 to 100 milliseconds. The
ready queue is treated as a circular queue. To implement RR scheduling
 We keep the ready queue as a FIFO queue of processes.
 New processes are added to the tail of the ready queue.

48
 The CPU scheduler picks the first process from the ready queue, sets a timer to interrupt after 1
time quantum, and dispatches the process.
 The process may have a CPU burst of less than 1 time quantum.
o In this case, the process itself will release the CPU voluntarily.
o The scheduler will then proceed to the next process in the ready queue.
 Otherwise, if the CPU burst of the currently running process is longer than 1 time quantum,
o The timer will go off and will cause an interrupt to the OS.
o A context switch will be executed, and the process will be put at the tail of the ready
queue.
o The CPU scheduler will then select the next process in the ready queue.
The average waiting time under the RR policy is often long. Consider the following set of processes that
arrive at time 0, with the length of the CPU burst given in milliseconds: (a time quantum of 4
milliseconds)
Burst Waiting Turnaround
Process Time Time Time
24 6 30
3 4 7
3 7 10
Average - 5.66 15.66
Using round-robin scheduling, we would schedule these processes according to the following chart:

49
DEQUE(Double Ended Queue):
A double-ended queue (dequeue, often abbreviated to deque, pronounced deck) generalizes a queue,
for which elements can be added to or removed from either the front (head) or back (tail).It is also often
called a head-tail linked list. Like an ordinary queue, a double-ended queue is a data structure it
supports the following operations: enq_front, enq_back, deq_front, deq_back, and empty. Dequeue can
be behave like a queue by using only enq_front and deq_front , and behaves like a stack by using only
enq_front and deq_rear.
The DeQueue is represented as follows.
DEQUE can be represented in two ways they are
1) Input restricted DEQUE(IRD)
2) output restricted DEQUE(ORD)
The output restricted DEQUE allows deletions from only one end and input restricted DEQUE allow
insertions at only one end. The DEQUE can be constructed in two ways they are
1) Using array
2)Using linked list
Operations in DEQUE
1. Insert element at back
2. Insert element at front
3. Remove element at front
4. Remove element at back

51
Applications of DEQUE:
1. The A-Steal algorithm implements task scheduling for several processors (multiprocessor
scheduling).
2. The processor gets the first element from the deque.
3. When one of the processor completes execution of its own threads it can steal a thread from
another processor.
4. It gets the last element from the deque of another processor and executes it.
Circular Queue:
Circular queue is a linear data structure. It follows FIFO principle. In circular queue the last node is
connected back to the first node to make a circle.
 Circular linked list fallow the First In First Out principle
 Elements are added at the rear end and the elements are deleted at front end of the queue
 Both the front and the rear pointers points to the beginning of the array.
 It is also called as “Ring buffer”.
 Items can inserted and deleted from a queue in O(1) time.
Circular Queue can be created in three ways they are
1. Using single linked list
2. Using double linked list
3. Using arrays

52
Representation of Circular Queue:
Let us consider a circular queue, which can hold maximum (MAX) of six elements. Initially the queue is
empty.
0
1
2
5
4
3
Q u e u e E mp t y
M A X = 6
F RO NT = RE A R = 0
C O U NT = 0
F R
C irc u lar Q u e u e
Now, insert 11 to the circular queue. Then circular queue status will be:
0
1
2
5
4
3
F RO NT = 0
RE A R = ( RE A R + 1) % 6 = 1
C O U NT = 1
F
1 1
R
Insert new elements 22, 33, 44 and 55 into the circular queue. The circular queue status is:
0
1
2
5
4
3
F RO NT = 0, RE A R = 5
RE A R = RE A R % 6 = 5
C O U NT = 5
F
1 1
R
2 2
4 4 3 3
5 5
Now, delete an element. The element deleted is the element at the front of the circular queue. So, 11 is
deleted. The circular queue status is as follows:

53
0
1
2
5
4
3
F RO NT = (F R O NT + 1) % 6 = 1
RE A R = 5
C O U NT = C O U NT - 1 = 4
F
R
2 2
4 4 3 3
5 5
Again, delete an element. The element to be deleted is always pointed to by the FRONT pointer. So, 22
is deleted. The circular queue status is as follows:
0
1
2
5
4
3
F RO NT = (F R O NT + 1) % 6 = 2
REA R = 5
C O U NT = C O U NT - 1 = 3
F
C irc ular Que ue
R
44 33
55
Again, insert another element 66 to the circular queue. The status of the circular queue is:
0
1
2
5
4
3
F RO NT = 2
RE A R = ( RE A R + 1) % 6 = 0
C O U NT = C O U NT + 1 = 4
R
4 4 3 3
5 5
6 6
F
Now, insert new elements 77 and 88 into the circular queue. The circular queue status is:

54
0
1
2
5
4
3
F RO NT = 2, REA R = 2
REA R = REA R % 6 = 2
CO U NT = 6
Circ ular Que ue
R
44 33
55
66
F
77
88
Now, if we insert an element to the circular queue, as COUNT = MAX we cannot add the element to
circular queue. So, the circular queue is full.

55
UNIT-3
LINKED LISTS
Linked lists and arrays are similar since they both store collections of data. Array is the most common
data structure used to store collections of elements. Arrays are convenient to declare and provide the easy
syntax to access any element by its index number. Once the array is set up, access to any element is
convenient and fast.
The disadvantages of arrays are:
• The size of the array is fixed. Most often this size is specified at compile time. This makes the
programmers to allocate arrays, which seems "large enough" than required.
• Inserting new elements at the front is potentially expensive because existing elements need to be
shifted over to make room.
• Deleting an element from an array is not possible. Linked lists have their own strengths and
weaknesses, but they happen to be strong where arrays are weak.
 Generally array's allocates the memory for all its elements in one block whereas linked lists use
an entirely different strategy. Linked lists allocate memory for each element separately and only
when necessary.
Linked List Concepts:
A linked list is a non-sequential collection of data items. It is a dynamic data structure. For every data
item in a linked list, there is an associated pointer that would give the memory location of the next data
item in the linked list. The data items in the linked list are not in consecutive memory locations. They
may be anywhere, but the accessing of these data items is easier as each data item contains the address of
the next data item.
Advantages of linked lists:
Linked lists have many advantages. Some of the very important advantages are:
1. Linked lists are dynamic data structures. i.e., they can grow or shrink during the execution of a
program.
2. Linked lists have efficient memory utilization. Here, memory is not preallocated. Memory is
allocated whenever it is required and it is de-allocated (removed) when it is no longer needed.
3. Insertion and Deletions are easier and efficient. Linked lists provide flexibility in inserting a data
item at a specified position and deletion of the data item from the given position.
4. Many complex applications can be easily carried out with linked lists.
Disadvantages of linked lists:
1. It consumes more space because every node requires a additional pointer to store address of the
next node.
2. Searching a particular element in list is difficult and also time consuming.
Types of Linked Lists:
Basically we can put linked lists into the following four items:
1. Single Linked List.
2. Double Linked List.
3. Circular Linked List.

56
4. Circular Double Linked List.
A single linked list is one in which all nodes are linked together in some sequential manner. Hence, it is
also called as linear linked list.
A double linked list is one in which all nodes are linked together by multiple links which helps in
accessing both the successor node (next node) and predecessor node (previous node) from any arbitrary
node within the list. Therefore each node in a double linked list has two link fields (pointers) to point to
the left node (previous) and the right node (next). This helps to traverse in forward direction and
backward direction.
A circular linked list is one, which has no beginning and no end. A single linked list can be made a
circular linked list by simply storing address of the very first node in the link field of the last node.
A circular double linked list is one, which has both the successor pointer and predecessor pointer in the
circular manner.

57
Applications of linked list:
1. Linked lists are used to represent and manipulate polynomial. Polynomials are expression containing
terms with non zero coefficient and exponents. For example: P(x) = a0 Xn + a1 Xn-1 + …… + an-1
X + an
2. Represent very large numbers and operations of the large number such as addition, multiplication and
division.
3. Linked lists are to implement stack, queue, trees and graphs. 4. Implement the symbol table in
compiler construction.
Single Linked List:
A linked list allocates space for each element separately in its own block of memory called a "node". The
list gets an overall structure by using pointers to connect all its nodes together like the links in a chain.
Each node contains two fields; a "data" field to store whatever element, and a "next" field which is a
pointer used to link to the next node. Each node is allocated in the heap using malloc(), so the node
memory continues to exist until it is explicitly de-allocated using free(). The front of the list is a pointer to
the “start” node.
A single linked list
The beginning of the linked list is stored in a "start" pointer which points to the first node. The first node
contains a pointer to the second node. The second node contains a pointer to the third node, ... and so on.
The last node in the list has its next field set to NULL to mark the end of the list. Code can access any
node in the list by starting at the start and following the next pointers.
The start pointer is an ordinary local pointer variable, so it is drawn separately on the left top to show that
it is in the stack. The list nodes are drawn on the right to show that they are allocated in the heap.
The basic operations in a single linked list are:
• Creation.
• Insertion.
• Deletion.
• Traversing.
Creating a node for Single Linked List:
Creating a singly linked list starts with creating a node. Sufficient memory has to be allocated for creating
a node. The information is stored in the memory.

58
Insertion of a Node:
One of the most primitive operations that can be done in a singly linked list is the insertion of a node.
Memory is to be allocated for the new node (in a similar way that is done while creating a list) before
reading the data. The new node will contain empty data field and empty next field. The data field of the
new node is then stored with the information read from the user. The next field of the new node is
assigned to NULL. The new node can then be inserted at three different places namely:
• Inserting a node at the beginning.
• Inserting a node at the end.
• Inserting a node at intermediate position.
 Inserting a node at the beginning.
 Inserting a node at the end:

59
 Inserting a node into the single linked list at a specified intermediate position other than
beginning and end.
Deletion of a node:
Another primitive operation that can be done in a singly linked list is the deletion of a node. Memory is
to be released for the node to be deleted. A node can be deleted from the list from three different places
namely.
• Deleting a node at the beginning.
• Deleting a node at the end.
• Deleting a node at intermediate position.
Deleting a node at the beginning:

60
Deleting a node at the end:
Deleting a node at Intermediate position:
The following steps are followed, to delete a node from an intermediate position in the list (List must
contain more than two node).
Traversal and displaying a list (Left to Right):
To display the information, you have to traverse (move) a linked list, node by node from the first node,
until the end of the list is reached.
Traversing a list involves the following steps:
• Assign the address of start pointer to a temp pointer.
• Display the information from the data field of each node. The function traverse () is used for traversing
and displaying the information stored in the list from left to right.
Source Code for creating and inserting the Implementation of Single Linked List:
class Item:
'''An Item that stores the data of a Linked List'''
'''Initializes the data of the Item'''
self._content = data # non-public data
self.next = None
class SingleLikedList:
'''Single Linked List Implementation'''
def __init__(self):
self._start = None

61
self._count = 0
def getItemAtIndex(self, pos):
if self._start == None:
return None
i = 0
cursor = self._start
while cursor != None:
if i == pos:
return cursor
i += 1
cursor = cursor.next
return None
def insert(self, data, pos = 0):
item = Item(data)
if self._start == None:
self._start = item
else:
cursor = self.getItemAtIndex(pos)
item.next = cursor.next
item.next = cursor.next
cursor.next = item
self._count += 1
def display(self):
cursor = self._start
while cursor != None:
print(cursor._content, end=' ')
cursor = cursor.next
l = SingleLikedList()
l.insert(10)
l.insert(20)

62
l.insert(30)
l.insert(40)
l.insert(50, 3)
l.display()
Source Code for creating , inserting ,deleting the Implementation of Single Linked List:
class Node(object):
def __init__(self, data=None, next_node=None):
self.data = data
self.next_node = next_node
def get_data(self):
return self.data
def get_next(self):
return self.next_node
def set_next(self, new_next):
self.next_node = new_next
class LinkedList(object):
def __init__(self, head=None):
self.head = head
def insert(self, data):
new_node = Node(data)
new_node.set_next(self.head)
self.head = new_node
def size(self):
current = self.head
count = 0
while current:
count += 1
current = current.get_next()
return count
def search(self, data):
current = self.head
found = False

63
while current and found is False:
if current.get_data() == data:
found = True
else:
if current is None:
raise ValueError("Data not in list")
return current
def delete(self, data):
current = self.head
previous = None
found = False
while current and found is False:
if current.get_data() == data:
found = True
else:
previous = current
if current is None:
raise ValueError("Data not in list")
if previous is None:
self.head = current.get_next()
else:
previous.set_next(current.get_next())
import sys
import os.path
sys.path.append(os.path.join(os.path.abspath(os.pardir), "/home/satya/PycharmProjects/DataStractures"))
from LinkedList2 import LinkedList
import unittest
class TestLinkedList(unittest.TestCase):
def setUp(self):
self.list = LinkedList()
def tearDown(self):

64
self.list = None
def test_insert(self):
self.list.insert("David")
self.assertTrue(self.list.head.get_data() == "David")
self.assertTrue(self.list.head.get_next() is None)
def test_insert_two(self):
self.list.insert("David")
self.list.insert("Thomas")
self.assertTrue(self.list.head.get_data() == "Thomas")
head_next = self.list.head.get_next()
self.assertTrue(head_next.get_data() == "David")
def test_nextNode(self):
self.list.insert("Jacob")
self.list.insert("Pallymay")
self.list.insert("Rasmus")
self.assertTrue(self.list.head.get_data() == "Rasmus")
head_next = self.list.head.get_next()
self.assertTrue(head_next.get_data() == "Pallymay")
last = head_next.get_next()
self.assertTrue(last.get_data() == "Jacob")
def test_positive_search(self):
found = self.list.search("Jacob")
self.assertTrue(found.get_data() == "Jacob")
found = self.list.search("Pallymay")
self.assertTrue(found.get_data() == "Pallymay")
def test_searchNone(self):
# make sure reg search works

65
with self.assertRaises(ValueError):
self.list.search("Vincent")
def test_delete(self):
# Delete the list head
self.list.delete("Rasmus")
self.assertTrue(self.list.head.get_data() == "Pallymay")
# Delete the list tail
self.list.delete("Jacob")
self.assertTrue(self.list.head.get_next() is None)
def test_delete_value_not_in_list(self):
self.list.delete("Sunny")
def test_delete_empty_list(self):
self.list.delete("Sunny")
def test_delete_next_reassignment(self):
self.list.insert("Cid")
self.list.delete("Pallymay")
self.list.delete("Cid")
self.assertTrue(self.list.head.next_node.get_data() == "Jacob")
class Node(object):
def __init__(self, data, next):
self.data = data

66
self.next = next
class SingleList(object):
head = None
tail = None
def show(self):
print "Showing list data:"
current_node = self.head
while current_node is not None:
print current_node.data, " -> ",
current_node = current_node.next
print None
def append(self, data):
node = Node(data, None)
if self.head is None:
self.head = self.tail = node
else:
self.tail.next = node
self.tail = node
def remove(self, node_value):
previous_node = None
if current_node.data == node_value:
# if this is the first node (head)
if previous_node is not None:
previous_node.next = current_node.next
else:
self.head = current_node.next
# needed for the next iteration
previous_node = current_node
s = SingleList()
s.append(31)
s.append(2)

67
s.append(3)
s.append(4)
s.show()
s.remove(31)
s.remove(3)
s.remove(2)
s.show()
Using a header node:
A header node is a special dummy node found at the front of the list. The use of header node is an
alternative to remove the first node in a list. For example, the picture below shows how the list with data
10, 20 and 30 would be represented using a linked list without and with a header node:
Note that if your linked lists do include a header node, there is no need for the special case code given
above for the remove operation; node n can never be the first node in the list, so there is no need to check
for that case. Similarly, having a header node can simplify the code that adds a node before a given node
n.
Note that if you do decide to use a header node, you must remember to initialize an empty list to contain
one (dummy) node, you must remember not to include the header node in the count of "real" nodes in the
list.
It is also useful when information other than that found in each node of the list is needed. For example,
imagine an application in which the number of items in a list is often calculated. In a standard linked list,
the list function to count the number of nodes has to traverse the entire list every time. However, if the
current length is maintained in a header node, that information can be obtained very quickly. 3.5. Array
based linked lists: Another alternative is to allocate the nodes in blocks. In fact, if you know the

68
maximum size of a list a head of time, you can pre-allocate the nodes in a single array. The result is a
hybrid structure – an array based linked list.
shows an example of null terminated single linked list where all the nodes are allocated contiguously in
an array.
Double Linked List: A double linked list is a two-way list in which all nodes will have two links. This
helps in accessing both successor node and predecessor node from the given node position. It provides bi-
directional traversing. Each node contains three fields:
 Left link.
 Data.
 Right link.
The left link points to the predecessor node and the right link points to the successor node. The data field
stores the required data.
Many applications require searching forward and backward thru nodes of a list. For example searching
for a name in a telephone directory would need forward and backward scanning thru a region of the whole
list.
The basic operations in a double linked list are:
 Creation.
 Insertion.
 Deletion.
 Traversing.
A double linked list is shown in figure

69
The beginning of the double linked list is stored in a "start" pointer which points to the first node. The
first node‟s left link and last node‟s right link is set to NULL.
Creating a node for Double Linked List:
Creating a double linked list starts with creating a node. Sufficient memory has to be allocated for
creating a node. The information is stored in the memory.
Double Linked List with 3 nodes:
Inserting a node at the beginning:
Inserting a node at the end:

70
Inserting a node at an intermediate position:

71
Traversal and displaying a list (Left to Right):
To display the information, you have to traverse the list, node by node from the first node, until the end
of the list is reached. The function traverse_left_right() is used for traversing and displaying the
information stored in the list from left to right.
Traversal and displaying a list (Right to Left):
To display the information from right to left, you have to traverse the list, node by node from the first
node, until the end of the list is reached. The function traverse_right_left() is used for traversing and
displaying the information stored in the list from right to left.
1.Source Code for the Implementation of Double Linked List:
class DList:
def __init__(self):
self.head = None
self.size = 0
def insert(self, data):
self.size += 1
self.head = Node(data)
else:
p = Node(data)
p.next = self.head
self.head.previous = p
self.head = p
def remove(self, index):
raise ValueError('Removing off an empty list')
if index < 0 or index >= self.size:
raise IndexError("Index is either negative or greater than the list size.")

72
current = self.head
if index == 0:
self.head = self.head.next
self.head.previous = None
else:
for _ in range(index -1):
current = current.next
p = current.next.next
if p is None:
current.next = None
else:
current.next = p
p.previous = current
def __sizeof__(self):
return self.size
def __repr__(self):
res = '[ '
current = self.head
while current is not None:
res += str(current.data)
res += ' '
current = current.next
res += ']'
return res
class Node:
if data is None:
raise ValueError('Node value cannot be None')
self.data = data
self.previous = None
self.next = None
Source Code for the Implementation of Double Linked List:
class Node(object):
def __init__(self, data, prev, next):

73
self.data = data
self.prev = prev
self.next = next
class DoubleList(object):
head = None
tail = None
def append(self, data):
new_node = Node(data, None, None)
self.head = self.tail = new_node
else:
new_node.prev = self.tail
new_node.next = None
self.tail.next = new_node
self.tail = new_node
def remove(self, node_value):
if current_node.data == node_value:
# if it's not the first element
if current_node.prev is not None:
current_node.prev.next = current_node.next
current_node.next.prev = current_node.prev
else:
# otherwise we have no prev (it's None), head is the next one, and prev becomes None
self.head = current_node.next
current_node.next.prev = None
def show(self):
print "Show list data:"

74
print current_node.prev.data if hasattr(current_node.prev, "data") else None,
print current_node.data,
print current_node.next.data if hasattr(current_node.next, "data") else None
print "*"*50
d = DoubleList()
d.append(5)
d.append(6)
d.append(50)
d.append(30)
d.show()
d.remove(50)
d.remove(5)
d.show()
Circular Single Linked List:
It is just a single linked list in which the link field of the last node points back to the address of the first
node. A circular linked list has no beginning and no end. It is necessary to establish a special pointer
called start pointer always pointing to the first node of the list. Circular linked lists are frequently used
instead of ordinary linked list because many operations are much easier to implement. In circular linked
list no null pointers are used, hence all pointers contain valid address.
Creating a circular single Linked List with „n‟ number of nodes:
The basic operations in a circular single linked list are:
• Creation.
• Insertion.
• Deletion.
• Traversing.

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Source Code for Circular Single Linked List:

76
from enum import Enum
class NodeConstants(Enum):
FRONT_NODE = 1
class Node:
def __init__(self, element=None, next_node=None):
self.element = element
self.next_node = next_node
def __str__(self):
if self.element:
return self.element.__str__()
else:
return 'Empty Node'
def __repr__(self):
return self.__str__()
class CircularLinkedList:
def __init__(self):
self.head = Node(element=NodeConstants.FRONT_NODE)
self.head.next_node = self.head
def size(self):
count = 0
current = self.head.next_node
while current != self.head:
count += 1
current = current.next_node
return count
def insert_front(self, data):
node = Node(element=data, next_node=self.head.next_node)
self.head.next_node = node
def insert_last(self, data):
current_node = self.head.next_node
while current_node.next_node != self.head:
current_node = current_node.next_node
node = Node(element=data, next_node=current_node.next_node)
current_node.next_node = node
def insert(self, data, position):

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if position == 0:
self.insert_front(data)
elif position == self.size():
self.insert_last(data)
else:
if 0 < position < self.size():
current_pos = 0
while current_pos < position - 1:
current_pos += 1
node = Node(data, current_node.next_node)
current_node.next_node = node
else:
raise IndexError
def remove_first(self):
self.head.next_node = self.head.next_node.next_node
def remove_last(self):
while current_node.next_node.next_node != self.head:
current_node.next_node = self.head
def remove(self, position):
if position == 0:
self.remove_first()
elif position == self.size():
self.remove_last()
else:
if 0 < position < self.size():
current_pos = 0
while current_pos < position - 1:
current_pos += 1
current_node.next_node = current_node.next_node.next_node

78
else:
raise IndexError
def fetch(self, position):
if 0 <= position < self.size():
current_pos = 0
while current_pos < position:
current_pos += 1
return current_node.element
else:
raise IndexError
import unittest
from random import randint
class TestCircularLinkedList(unittest.TestCase):
names = ['Bob Belcher',
'Linda Belcher',
'Tina Belcher',
'Gene Belcher',
'Louise Belcher']
def test_init(self):
dll = CircularLinkedList()
self.assertIsNotNone(dll.head)
self.assertEqual(dll.size(), 0)
def test_insert_front(self):
for name in TestCircularLinkedList.names:
dll.insert_front(name)
self.assertEqual(dll.fetch(0), TestCircularLinkedList.names[4])
def test_insert_last(self):

79
dll.insert_last(name)
for i in range(len(TestCircularLinkedList.names) - 1):
self.assertEqual(dll.fetch(i), TestCircularLinkedList.names[i])
def test_insert(self):
pos = randint(0, len(TestCircularLinkedList.names) - 1)
dll.insert('Teddy', pos)
self.assertEqual(dll.fetch(pos), 'Teddy')
def test_remove_first(self):
for i in range(dll.size(), 0, -1):
self.assertEqual(dll.size(), i)
dll.remove_first()
def test_remove_last(self):
for i in range(dll.size(), 0, -1):
self.assertEqual(dll.size(), i)
dll.remove_last()
def test_remove(self):
dll.remove(1)
self.assertEqual(dll.fetch(0), 'Bob Belcher')
self.assertEqual(dll.fetch(1), 'Tina Belcher')
self.assertEqual(dll.fetch(2), 'Gene Belcher')
self.assertEqual(dll.fetch(3), 'Louise Belcher')

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if __name__ == '__main__':
unittest.main()
Circular Double Linked List:
A circular double linked list has both successor pointer and predecessor pointer in circular manner. The
objective behind considering circular double linked list is to simplify the insertion and deletion operations
performed on double linked list. In circular double linked list the right link of the right most node points
back to the start node and left link of the first node points to the last node.
A circular double linked list is shown in figure
Creating a Circular Double Linked List with „n‟ number of nodes
The basic operations in a circular double linked list are:
• Creation.
• Insertion.
• Deletion.
• Traversing.

81
Inserting a node at an intermediate position:

82
Comparison of Linked List Variations:
The major disadvantage of doubly linked lists (over singly linked lists) is that they require more space
(every node has two pointer fields instead of one). Also, the code to manipulate doubly linked lists needs
to maintain the prev fields as well as the next fields; the more fields that have to be maintained, the more
chance there is for errors.
The major advantage of doubly linked lists is that they make some operations (like the removal of a given
node, or a right-to-left traversal of the list) more efficient.
The major advantage of circular lists (over non-circular lists) is that they eliminate some extra-case code
for some operations (like deleting last node). Also, some applications lead naturally to circular list
representations. For example, a computer network might best be modeled using a circular list.
Polynomials:
A polynomial is of the form: i n i ∑ ci
Where, ci is the coefficient of the ith term and
n is the degree of the polynomial
Some examples are:
5x2 + 3x + 1
12x3 – 4x
5x4 – 8x3 + 2x2 + 4x1 + 9x0

83
It is not necessary to write terms of the polynomials in decreasing order of degree. In other words the two
polynomials 1 + x and x + 1 are equivalent.
The computer implementation requires implementing polynomials as a list of pairs of coefficient and
exponent. Each of these pairs will constitute a structure, so a polynomial will be represented as a list of
structures.
A linked list structure that represents polynomials 5x4 – 8x3 + 2x2 + 4x1 + 9x0
Addition of Polynomials:
To add two polynomials we need to scan them once. If we find terms with the same exponent in the two
polynomials, then we add the coefficients; otherwise, we copy the term of larger exponent into the sum
and go on. When we reach at the end of one of the polynomial, then remaining part of the other is copied
into the sum.
To add two polynomials follow the following steps:
• Read two polynomials
• Add them.
• Display the resultant polynomial.

84
UNIT – IV NON LINEAR DATA STRUCTURES
Trees Basic Concepts:
A tree is a non-empty set one element of which is designated the root of the tree while the remaining
elements are partitioned into non-empty sets each of which is a sub-tree of the root.
A tree T is a set of nodes storing elements such that the nodes have a parent-child relationship that
satisfies the following
• If T is not empty, T has a special tree called the root that has no parent.
• Each node v of T different than the root has a unique parent node w; each node with parent w is a
child of w.
Tree nodes have many useful properties. The depth of a node is the length of the path (or the number of
edges) from the root to that node. The height of a node is the longest path from that node to its leaves.
The height of a tree is the height of the root. A leaf node has no children -- its only path is up to its
parent.
Binary Tree:
In a binary tree, each node can have at most two children. A binary tree is either empty or consists of a
node called the root together with two binary trees called the left subtree and the right subtree.

85
Tree Terminology:
Leaf node
A node with no children is called a leaf (or external node). A node which is not a leaf is called an internal
node.
Path: A sequence of nodes n1, n2, . . ., nk, such that ni is the parent of ni + 1 for i = 1, 2,. . ., k - 1. The
length of a path is 1 less than the number of nodes on the path. Thus there is a path of length zero from a
node to itself.
Siblings: The children of the same parent are called siblings.
Ancestor and Descendent If there is a path from node A to node B, then A is called an ancestor of B
and B is called a descendent of A.
Subtree: Any node of a tree, with all of its descendants is a subtree.
Level: The level of the node refers to its distance from the root. The root of the tree has level O, and the
level of any other node in the tree is one more than the level of its parent.
The maximum number of nodes at any level is 2n
.
Height:The maximum level in a tree determines its height. The height of a node in a tree is the length of a
longest path from the node to a leaf. The term depth is also used to denote height of the tree.
Depth:The depth of a node is the number of nodes along the path from the root to that node.
Assigning level numbers and Numbering of nodes for a binary tree: The nodes of a binary tree can be
numbered in a natural way, level by level, left to right. The nodes of a complete binary tree can be
numbered so that the root is assigned the number 1, a left child is assigned twice the number assigned its
parent, and a right child is assigned one more than twice the number assigned its parent.

86
Properties of Binary Trees:
Some of the important properties of a binary tree are as follows:
1. If h = height of a binary tree, then
a. Maximum number of leaves = 2h
b. Maximum number of nodes = 2h + 1
- 1
2. If a binary tree contains m nodes at level l, it contains at most 2m nodes at level l + 1.
3. Since a binary tree can contain at most one node at level 0 (the root), it can contain at most 2l
node at level l.
4. The total number of edges in a full binary tree with n node is n - 1.
Strictly Binary tree:
If every non-leaf node in a binary tree has nonempty left and right subtrees, the tree is termed a strictly
binary tree. Thus the tree of figure 7.2.3(a) is strictly binary. A strictly binary tree with n leaves always
contains 2n - 1 nodes.
Full Binary Tree:
A full binary tree of height h has all its leaves at level h. Alternatively; All non leaf nodes of a full binary
tree have two children, and the leaf nodes have no children.
A full binary tree with height h has 2h + 1
- 1 nodes. A full binary tree of height h is a strictly binary tree all
of whose leaves are at level h.

87
For example, a full binary tree of height 3 contains 23+1
– 1 = 15 nodes.
Complete Binary Tree:
A binary tree with n nodes is said to be complete if it contains all the first n nodes of the above
numbering scheme.
A complete binary tree of height h looks like a full binary tree down to level h-1, and the level h is filled
from left to right.
Perfect Binary Tree:
A Binary tree is Perfect Binary Tree in which all internal nodes have two children and all leaves are at
same level.
Following are examples of Perfect Binary Trees.
18
/
15 30
/ /
40 50 100 40

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18
/
15 30
A Perfect Binary Tree of height h (where height is number of nodes on path from root to leaf) has 2h
– 1
node.
Example of Perfect binary tree is ancestors in family. Keep a person at root, parents as children, parents
of parents as their children.
Balanced Binary Tree:
A binary tree is balanced if height of the tree is O(Log n) where n is number of nodes. For Example, AVL
tree maintain O(Log n) height by making sure that the difference between heights of left and right
subtrees is 1. Red-Black trees maintain O(Log n) height by making sure that the number of Black nodes
on every root to leaf paths are same and there are no adjacent red nodes. Balanced Binary Search trees are
performance wise good as they provide O(log n) time for search, insert and delete.
Representation of Binary Trees:
1. Array Representation of Binary Tree
2. Pointer-based.
Array Representation of Binary Tree:
A single array can be used to represent a binary tree.
For these nodes are numbered / indexed according to a scheme giving 0 to root. Then all the
nodes are numbered from left to right level by level from top to bottom. Empty nodes are also
numbered. Then each node having an index i is put into the array as its ith
element.
In the figure shown below the nodes of binary tree are numbered according to the given scheme.

89
The figure shows how a binary tree is represented as an array. The root 3 is the 0th
element while its
leftchild 5 is the 1 st element of the array. Node 6 does not have any child so its children i.e. 7 th
and 8 th
element of the array are shown as a Null value.
It is found that if n is the number or index of a node, then its left child occurs at (2n + 1)th
position and
right child at (2n + 2) th
position of the array. If any node does not have any of its child, then null value is
stored at the corresponding index of the array.
The following program implements the above binary tree in an array form. And then traverses
the tree in inorder traversal.
# Python implementation to construct a Binary Tree from
# parent array
# A node structure
class Node:
# A utility function to create a new node
def __init__(self, key):
self.key = key
self.left = None
self.right = None
""" Creates a node with key as 'i'. If i is root,then
it changes root. If parent of i is not created, then
it creates parent first
"""
def createNode(parent, i, created, root):
# If this node is already created
if created[i] is not None:
return
# Create a new node and set created[i]
created[i] = Node(i)
# If 'i' is root, change root pointer and return

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