Data Structure and Algorithms (DSA) with Python

This is the summer internship report. This report is the approved 'sample' outline for a summer internship mandatory submission. Titled, "Data Structure with Algorithms with Python".

1. A
Training report on
“DSA with Python”
submitted
in partial fulfilment
for the award of the Degree of
Bachelor of Technology
In Department of Computer Science & Engineering
(With specialisation in Computer Science & Engineering)
Submitted By:
Student Name
Roll No.: 22EJExxxxxx
Submitted To:
Mr. Prof. Name
(HOD of CSE)
Department of Computer Science & Engineering
Jaipur Engineering College, Kukas
Rajasthan Technical University
November, 2024

2. JAIPUR ENGINEERING COLLEGE
(Approved by AICTE, New Delhi &
Affiliated to Rajasthan Technical University, Kota)
Internal Guide Certificate
This is to certify that Student Name student of B. Tech (Computer Science) academic year
(2022-2026) at JAIPUR ENGINEERING COLLEGE has completed Summer Training
Report entitled. “DSA with Python”. The training has been completed in V Semester
course and for partially fulfilling the requirements of Rajasthan Technical University,
Kota. The Training Report has been completed under the guidance of Mr. Prof. Name,
Professor (CS Department) of JEC and is as per norms and guidelines provided.
Mr. Prof. Name
(CSE Dept. Professor)
Mr. Prof. Name
(HOD of CSE Dept.)
SP-43, RIICO Industrial Area, Kukas, Jaipur (Rajasthan)
Phone Number: 01426-511241/42/43, Fax: [phone]
www.jeckukas.org.in
ii

3. CERTIFICATE

4. ACKNOWLEDGEMENT
I express my sincere thanks to Professor Mr. Prof. Name, of the Department of
Computer Science & Engineering, for guiding me right form the inception till the
successful completion of the training. I sincerely acknowledge him for extending their
valuable guidance, support for literature, critical reviews of training and this report and
above all for the moral support he provided me at all stages of training. I would also like to
thank the supporting staff for their help and cooperation throughout my training.
Student Name: Student Name
Roll Number: 22EJxxxxxx
iv

5. ABSTRACT
In our exploration of DSA with Python, we have embarked on a comprehensive journey
through the intricate world of data structures and algorithms. This journey has taken us from
the foundational elements such as linked lists, stacks, and queues, to more advanced
structures like hash tables, trees, and graphs. Each data structure unveiled its unique
strengths and applications, demonstrating the elegance and efficiency that can be achieved
with proper implementation.
We have delved into the mastery of various algorithms, from sorting techniques like bubble
sort, merge sort, and quick sort, to search algorithms such as linear and binary search. We
explored the realms of graph traversal with depth-first and breadth-first searches, and
embraced the power of optimization with Dijkstra's and Prim's algorithms. Through the lens
of divide and conquer, greedy strategies, and backtracking, we learned to tackle complex
problems with clarity and precision.
v

6. TABLE OF CONTENTS
TITLE PAGE (i)
INTERNAL GUID CERTIFICATE (ii)
CERTIFICATE (iii)
ACKNOWLEDGEMENT (iv)
ABSTRACT (v)
Sr. No. Chapter Pg. No.
1 INTRODUCTION WITH PYTHON 1
1.1 Python Basics 1
1.2 Control Flow 2
1.3 Functions and Modules 4
1.4 OOP 5
1.5 File Handling 9
2 PYTHON DATA ARRANGEMENT FORMATS 12
2.1 List and Tuples 12
2.2 Dictionaries and Sets 14
2.3 Arrays 16
3 DATA STRUCTURE 19
3.1 Introduction 19
3.2 Linked List 20
3.3 Stack 21
3.4 Queue 23
3.5 Hash 27
3.6 Tree 28

7. 3.7 Graph 30
3.8 Heap 31
4 ALGORITHMS 35
4.1 Introduction 35
4.2 Time and Space Complexity 35
4.3 Search Algorithms 37
4.4 Divide and Conquer 40
4.5 Sorting Algorithms 40
4.6 Greedy Algorithms 43
4.7 Backtracking 45
4.7 Graph Algorithms 47
CONCLUSION 49
REFERENCE 50
vii

8. 1
Chapter – 1
INTRODUCTION TO PYTHON
1.1 Python Basics
Python is a high-level, interpreted programming language known for its readability and
simplicity. Its syntax allows developers to write code that is easy to understand and
maintain. Python's versatility and wide range of libraries make it a popular choice for
various applications, including web development, data analysis, artificial intelligence,
scientific computing, and more.
Python is a high-level, interpreted programming language known for its readability and
simplicity. Its syntax allows developers to write code that is easy to understand and
maintain. Python's versatility and wide range of libraries make it a popular choice for
various applications, including web development, data analysis, artificial intelligence,
scientific computing, and more.
1.1.1 Variables and Data Types
In Python, variables are used to store data values. Unlike some other programming
languages, you don't need to declare a variable type in Python. Here are a few
common data types you’ll encounter:
Integers: These are whole numbers, both positive and negative. For
example:
x = 5
y = -3
Floats: These are decimal point numbers. For example:
pi = 3.14
temperature = -5.0
Strings: A sequence of characters enclosed in quotes. Single (‘ ’) or
double (“ ”) quotes can be used. For example:
name = "Alice"
greeting = 'Hello'
Booleans: These represent True or False values. For example:
is_valid = True
is_completed = False
1.1.2 Basic Operations
Python supports a wide range of operations that can be performed on its various data
types. Here’s a brief overview:
Arithmetic Operations:
Addition (+): result = x + y
Subtraction (-): result = x - y

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Multiplication (*): result = x * y
Division (/): result = x / y
Modulus (%): remainder = x % y
Comparison Operations:
Equal to (==): is_equal = (x == y)
Not equal to (!=): is_not_equal = (x != y)
Greater than (>): is_greater = (x > y)
Less than (<): is_lesser = (x < y)
Logical Operations:
And (and): result = (x > 0) and (y < 10)
Or (or): result = (x > 0) or (y < 0)
Not (not): result = not (x > y)
1.1.3 Input and Output
Taking input from users and displaying output is fundamental in any programming
language. Here’s how it’s done in Python:
Input: The input() function is used to take input from the user. It always
returns the input as a string. For example:
user_input = input("Enter your name: ")
print("Hello, " + user_input + "!")
Output: The print() function is used to display output. You can print text
and variables. For example:
print("The value of x is:", x)
print("Hello, World!")
1.2 Control Flow
Control flow is the order in which the individual statements, instructions, or function calls
of a program are executed or evaluated. In Python, control flow is managed through
conditionals, loops, and functions. By controlling the flow of the program, you can make
decisions and execute code based on specific conditions.
1.2.1 Conditional Statements
Conditional statements allow you to execute certain parts of code based on whether
a condition is true or false. The most commonly used conditional statements in
Python are if, elif, and else.
Syntax:
if condition:
# Code to execute if the condition is true
elif another_condition:

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# Code to execute if another_condition is true
else:
# Code to execute if none of the conditions are true
Example:
x = 10
if x > 0:
print("x is positive")
elif x == 0:
print("x is zero")
else:
print("x is negative")
1.2.2 Loops
Loops are used to repeatedly execute a block of code as long as a condition is true.
For Loop: A for loop is used for iterating over a sequence (such as a list,
tuple, dictionary, set, or string).
Syntax:
for item in iterable:
# Code to execute for each item
Example:
fruits = ["apple", "banana", "cherry"]
for fruit in fruits:
print(fruit)
While Loop: A while loop will repeatedly execute a block of code as long as
the condition is true.
Syntax:
while condition:
# Code to execute as long as the condition is
true
Example:
i = 1
while i < 6:
print(i)
i += 1
1.2.3 Exception Handling
Exception handling is a way to handle errors gracefully in your program. Instead of
your program crashing when an error occurs, you can use exception handling to
catch the error and do something about it.

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Try-Except Block: The try block lets you test a block of code for errors, the
except block lets you handle the error.
Syntax:
try:
# Code that might raise an exception
except SomeException as e:
# Code to handle the exception
else:
# Code to execute if no exception is raised
finally:
# Code that will always execute, regardless of
an exception
Example:
try:
x = 10 / 0
except ZeroDivisionError as e:
print("Cannot divide by zero:", e)
else:
print("No error occurred")
finally:
print("Execution completed")
1.3 Functions and Modules
Functions and modules are key components in Python that help in organizing and reusing
code. Let’s explore them in detail.
1.3.1 Defining Functions
Functions in Python allow you to encapsulate a block of code that performs a
specific task, making it reusable and easier to manage.
Syntax:
def function_name(parameters):
# code block
return result
Example:
def greet(name):
return f"Hello, {name}!"
print(greet("Alice"))

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1.3.2 Importing Modules
Modules are files containing Python code (variables, functions, classes) that you can
import and use in your programs. Python has a rich standard library, and you can
also create your own modules.
Syntax:
import module_name
Example:
import math
print(math.sqrt(16)) # Outputs: 4.0
Importing Specific Items:
from math import sqrt
print(sqrt(25)) # Outputs: 5.0
Importing with Aliases:
import math as m
print(m.sqrt(36)) # Outputs: 6.0
1.3.3 Creating Your Own Modules
Creating your own module is straightforward. You simply write Python code in a file
and save it with a .py extension. Then, you can import this file as a module.
Create a file my_module.py:
def add(a, b):
return a + b
def subtract(a, b):
return a – b
Create a main program to use the module:
import my_module
print(my_module.add(5, 3)) # Outputs: 8
print(my_module.subtract(5, 3)) # Outputs: 2
1.4 OOP
Object-Oriented Programming (OOP) is a programming paradigm that uses objects and
classes to create models based on the real world. It’s useful for organizing complex
programs, improving reusability, and enhancing code readability. By encapsulating data
and functions into objects, OOP promotes modularity and helps in managing the
complexity of software systems by allowing for code to be more easily maintained and
extended.

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1.4.1 Classes and Objects
Classes are blueprints for creating objects. A class defines a set of attributes and
methods that the created objects will have. Objects are instances of a class.
Syntax:
class ClassName:
# Attributes and methods
def __init__(self, attribute1, attribute2):
self.attribute1 = attribute1
self.attribute2 = attribute2
def method(self):
# Method definition
pass
Example:
class Dog:
def __init__(self, name, breed):
self.name = name
self.breed = breed
def bark(self):
return f"{self.name} says woof!"

my_dog = Dog("Buddy", "Golden Retriever")
print(my_dog.bark()) # Outputs: Buddy says woof!
1.4.2 Inheritance
Inheritance allows one class (child class) to inherit attributes and methods from
another class (parent class). This promotes code reusability and hierarchical
classification, making it easier to manage and understand the code structure. It also
supports polymorphism, enabling objects to be treated as instances of their parent
class, which enhances flexibility. Moreover, inheritance helps in reducing
redundancy by allowing common functionality to be defined in a base class.

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Syntax:
class ParentClass:
# Parent class definition
class ChildClass(ParentClass):
# Child class definition
Example:
class Animal:
def __init__(self, name):
self.name = name
def speak(self):
pass
class Dog(Animal):
def speak(self):
return f"{self.name} says woof!"
class Cat(Animal):
def speak(self):
return f"{self.name} says meow!"
dog = Dog("Buddy")
cat = Cat("Whiskers")
print(dog.speak()) # Outputs: Buddy says woof!
print(cat.speak()) # Outputs: Whiskers says meow!
1.4.3 Encapsulation
Encapsulation is the process of wrapping data (attributes) and methods into a single
unit, a class. It also involves restricting access to certain details of an object.
class BankAccount:
def __init__(self, balance):
self.__balance = balance # Private attribute
def deposit(self, amount):

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self.__balance += amount
def withdraw(self, amount):
if amount <= self.__balance:
self.__balance -= amount
else:
print("Insufficient funds")
def get_balance(self):
return self.__balance
account = BankAccount(1000)
account.deposit(500)
account.withdraw(200)
print(account.get_balance()) # Outputs: 1300
1.4.4 Polymorphism
Polymorphism allows methods to do different things based on the object it is acting
upon, even though they share the same name.
class Bird:
def fly(self):
return "Birds can fly"
class Penguin(Bird):
def fly(self):
return "Penguins cannot fly"
bird = Bird()
penguin = Penguin()
print(bird.fly()) # Outputs: Birds can fly
print(penguin.fly()) # Outputs: Penguins cannot fly

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1.5 File Handling
File handling is an essential part of programming that allows you to read from, write to, and
manipulate files on your system. It’s crucial for various applications, such as data analysis,
logging, and more.
1.5.1 Reading Files
Opening a File: To read a file, you first need to open it. Python provides the
open() function for this purpose. You can specify the mode in which you
want to open the file—'r' for reading, 'w' for writing, 'a' for
appending, and 'b' for binary mode.
Syntax:
file = open('filename', 'mode')
Example:
file = open('example.txt', 'r')
content = file.read()
print(content)
file.close()
Using the with statement: The with statement is often used for file
operations because it ensures that the file is properly closed after its suite
finishes.
with open('example.txt', 'r') as file:
content = file.read()
print(content)
1.5.2 Writing Files
Writing to a File: To write to a file, you open it in write ('w') or append
('a') mode. If the file doesn't exist, it will be created.
with open('example.txt', 'w') as file:
file.write("Hello, World!")

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Appending to a File: Appending adds new content to the end of the file
without overwriting existing content.
with open('example.txt', 'a') as file:
file.write("\nThis is an appended line.")
1.5.3 Working with CSV Files
CSV (Comma-Separated Values) files are commonly used for storing tabular data.
Python provides the csv module to work with CSV files.
Reading CSV Files:
import csv
with open('data.csv', 'r') as file:
reader = csv.reader(file)
for row in reader:
print(row)
Writing to CSV Files:
import csv
with open('data.csv', 'w', newline='') as file:
writer = csv.writer(file)
writer.writerow(['Name', 'Age', 'City'])
writer.writerow(['Alice', 25, 'New York'])
writer.writerow(['Bob', 30, 'San Francisco'])
Reading CSV Files into a Dictionary: Sometimes, it’s more convenient to
read CSV data into a dictionary.
import csv
with open('data.csv', 'r') as file:
reader = csv.DictReader(file)

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for row in reader:
print(dict(row))
Writing to CSV Files from a Dictionary:
import csv
data = [
{'Name': 'Alice', 'Age': 25, 'City': 'New
York'},
{'Name': 'Bob', 'Age': 30, 'City': 'San
Francisco'}
]
with open('data.csv', 'w', newline='') as file:
fieldnames = ['Name', 'Age', 'City']
writer = csv.DictWriter(file,
fieldnames=fieldnames)

writer.writeheader()
for row in data:
writer.writerow(row)

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Chapter – 2
PYTHON DATA ARRANGEMENT FORMATS
2.1 Lists and Tuples
Lists and tuples in Python are both used to store collections of items, but they serve slightly
different purposes due to their inherent properties. Lists, which are enclosed in square
brackets, are mutable, meaning that the contents can be changed after they are created. This
flexibility allows for modifications such as adding, removing, or altering elements within
the list. This makes lists particularly useful for scenarios where the data set needs to be
dynamic or updated frequently. For example, lists are often used to store sequences of data
that will change, such as user inputs, configurations that may need to be adjusted, or data
that evolves over the course of a program's execution. The ability to change the list makes it
versatile for a wide range of applications, from simple data aggregation to complex data
manipulation tasks in various domains, including web development, data analysis, and
machine learning.
On the other hand, tuples are enclosed in parentheses and are immutable, meaning once they
are created, their contents cannot be altered. This immutability provides a form of integrity
and security, ensuring that the data cannot be changed accidentally or intentionally, which
can be crucial for maintaining consistent data states. Tuples are often used to represent fixed
collections of items, such as coordinates, RGB colour values, or any data set that should
remain constant throughout the program. The immutability of tuples can lead to more
predictable and bug-free code, as developers can be certain that the data will not be altered.
Additionally, because they are immutable, tuples can be used as keys in dictionaries, which
require immutable types. This makes tuples suitable for scenarios where constant and
unchanging data is necessary, enhancing both the readability and reliability of the code.
2.1.1 Creating Lists and Tuples
Creating a List: A list is created by placing all the items (elements) inside
square brackets [], separated by commas.
my_list = [1, 2, 3, 4, 5]
Creating a Tuple: A tuple is created by placing all the items (elements)
inside parentheses (), separated by commas.
my_tuple = (1, 2, 3, 4, 5)

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2.1.2 Basic Operations
Both lists and tuples support common operations such as indexing, slicing, and
iteration.
Indexing: You can access individual elements using an index. Indexing
starts from 0.
print(my_list[0]) # Outputs: 1
print(my_tuple[1]) # Outputs: 2
Slicing: You can access a range of elements using slicing.
print(my_list[1:3]) # Outputs: [2, 3]
print(my_tuple[:4]) # Outputs: (1, 2, 3, 4)
Iteration: You can loop through the elements using a for loop.
for item in my_list:
print(item)
for item in my_tuple:
print(item)
2.1.3 List and Tuple Methods
Lists have several built-in methods that allow modification. Tuples, being
immutable, have fewer methods.
List Methods:
append(): Adds an item to the end of the list.
my_list.append(6)
print(my_list) # Outputs: [1, 2, 3, 4, 5, 6]
remove(): Removes the first occurrence of an item.
my_list.remove(3)
print(my_list) # Outputs: [1, 2, 4, 5, 6]
insert(): Inserts an item at a specified position.
my_list.insert(2, 'a')
print(my_list) # Outputs: [1, 2, 'a', 4, 5, 6]

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pop(): Removes and returns the item at a given position. If no index is
specified, removes and returns the last item.
my_list.pop()
print(my_list) # Outputs: [1, 2, 'a', 4, 5]
Tuple Methods:
count(): Returns the number of times a specified value occurs in a tuple.
print(my_tuple.count(2)) # Outputs: 1
index(): Searches the tuple for a specified value and returns the position
of where it was found.
print(my_tuple.index(3)) # Outputs: 2
2.2 Dictionaries and Sets
Dictionaries and sets are fundamental data structures in Python, each with distinct
characteristics that make them suitable for different scenarios, especially in object-oriented
programming and machine learning contexts. Dictionaries, often called associative arrays
or hash maps in other programming languages, store data as key-value pairs, allowing for
efficient data retrieval based on unique keys. This makes dictionaries particularly useful for
tasks where quick lookup, insertion, and deletion of data are critical. For instance,
dictionaries are invaluable in machine learning for managing datasets, where feature names
(keys) are mapped to their values, or for storing model parameters and configurations. Their
ability to handle complex data associations dynamically makes them a versatile tool in both
data preprocessing and algorithm implementation.
Sets, on the other hand, are collections of unique elements, which makes them ideal for
operations involving membership tests, duplicates removal, and mathematical operations
like unions, intersections, and differences. In the context of machine learning, sets are often
used to handle unique elements such as in feature selection processes, where ensuring that
each feature is considered only once is crucial. Additionally, sets are employed to efficiently
manage and analyze datasets by quickly identifying and eliminating duplicates, thus
maintaining data integrity. Both dictionaries and sets contribute significantly to the
robustness and efficiency of programming solutions, providing the tools needed to handle
various data-centric tasks with precision and ease. Their distinct properties—dictionaries
with their key-value mappings and sets with their uniqueness constraints—complement
each other and enhance the flexibility and performance of Python applications in both
everyday programming and specialized fields like machine learning.

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2.2.1 Creating Dictionaries and Sets
Dictionaries: Dictionaries are collections of key-value pairs. Each key maps
to a specific value, and keys must be unique. Dictionaries are created using
curly braces {} with a colon separating keys and values.
my_dict = {'name': 'Alice', 'age': 25, 'city': 'New
York'}
Sets: Sets are unordered collections of unique elements. Sets are created
using curly braces {} or the set() function.
my_set = {1, 2, 3, 4, 5}
2.2.2 Basic Operations
Dictionaries:
Accessing Values: You can access dictionary values using their keys.
print(my_dict['name']) # Outputs: Alice
Adding/Updating Elements: You can add new key-value pairs or
update existing ones.
my_dict['age'] = 26 # Updates age to 26
Removing Elements: You can remove elements using the del statement
or pop() method.
del my_dict['city'] # Removes the key 'city'
Sets:
Adding Elements: Use the add() method to add elements to a set.
my_set.add(6) # Adds 6 to the set
Removing Elements: Use the remove() or discard() method to remove
elements.
my_set.remove(3) # Removes 3 from the set
Set Operations: Sets support mathematical operations like union,
intersection, and difference.
another_set = {4, 5, 6, 7}
union_set = my_set.union(another_set) # {1, 2,
4, 5, 6, 7}

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2.2.3 Dictionary and Set Methods
Dictionary Methods:
keys(): Returns a view object that displays a list of all the keys in the
dictionary.
print(my_dict.keys()) # Outputs:
dict_keys(['name', 'age'])
values(): Returns a view object that displays a list of all the values in the
dictionary.
print(my_dict.values()) # Outputs:
dict_values(['Alice', 26])
items(): Returns a view object that displays a list of dictionary's key-
value tuple pairs.
print(my_dict.items()) # Outputs:
dict_items([('name', 'Alice'), ('age', 26)])
Set Methods:
union(): Returns a new set containing all elements from both sets.
union_set = my_set.union(another_set) # {1, 2,
4, 5, 6, 7}
intersection(): Returns a new set containing only elements that are
common to both sets.
intersection_set = my_set.intersection
(another_set) # {4, 5}
difference(): Returns a new set containing elements that are in the first
set but not in the second.
difference_set = my_set.difference(another_set)
# {1, 2}
2.3 Arrays
Arrays are one of the most fundamental data structures in computer science. They provide a
way to store a fixed-size sequential collection of elements of the same type. Arrays are
widely used because they offer efficient access to elements by their index and support
various operations that are foundational for many algorithms.

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2.3.1 Creating Arrays in Python
The closest native data structure to arrays is the list, but for more efficiency with
numerical operations, the array module or NumPy arrays are often used.
Using the array Module:
import array
# Creating an array of integers
arr = array.array('i', [1, 2, 3, 4, 5])
print(arr) # Outputs: array('i', [1, 2, 3, 4, 5])
# Accessing elements by index
print(arr[0]) # Outputs: 1
# Modifying elements
arr[1] = 10
print(arr) # Outputs: array('i', [1, 10, 3, 4, 5])
Using NumPy Arrays:
import numpy as np
# Creating a NumPy array
arr = np.array([1, 2, 3, 4, 5])
print(arr) # Outputs: [1 2 3 4 5]
# Accessing elements by index
print(arr[0]) # Outputs: 1
# Modifying elements
arr[1] = 10
print(arr) # Outputs: [ 1 10 3 4 5]

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# Performing operations on arrays
print(arr + 2) # Outputs: [ 3 12 5 6 7]
print(arr * 2) # Outputs: [ 2 20 6 8 10]
2.3.2 Applications of Arrays in Algorithms
Sorting Algorithms: Arrays are often used in sorting algorithms such as
Quick Sort, Merge Sort, and Bubble Sort. Sorting algorithms typically
operate on arrays due to their efficient access and modification capabilities.
Searching Algorithms: Arrays are used in searching algorithms such as
Linear Search and Binary Search. Arrays provide a straightforward way to
store data sequentially, making it easier to search through them.
Dynamic Programming: Arrays are crucial in dynamic programming to
store intermediate results and avoid redundant calculations. This helps in
optimizing the time complexity of algorithms.

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Chapter – 3
DATA STRUCTURE
3.1 Introduction
Data structures are specialized formats for organizing, processing, and storing data.
3.1.1 Primitive Data Structures:
These are basic structures that hold single values. They are the simplest form of data
storage.
Integers
Floats
Characters
Booleans
3.1.2 Non-Primitive Data Structures:
These are more complex structures that can hold multiple values and can be used to
store collections of data.
Linear: These data structures store data in a linear or sequential order. Every
element has a single predecessor and a single successor (except the first and
last elements).
Arrays
Linked List
Stacks
Queues
Non-Linear Data Structures: These data structures store data in a
hierarchical manner and do not necessarily follow a sequential order. Each
element can be connected to multiple elements, reflecting complex
relationships.
Trees
Graphs

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3.2 Linked List
A linked list is a linear data structure where elements are stored in nodes. Each node
contains two parts: data and a reference (or link) to the next node in the sequence.
Singly Linked List
Doubly Linked List
Circular Linked List
3.2.1 Singly Linked List
Each node in a singly linked list points to the next node. The last node points to null,
indicating the end of the list.
class Node:
def __init__(self, data):
self.data = data
self.next = None
class SinglyLinkedList:
def __init__(self):
self.head = None
3.2.2 Doubly Linked List
Each node has two references: one to the next node and another to the previous node.
This allows traversal in both directions.
class Node:
def __init__(self, data):
self.data = data
self.next = None
self.prev = None
class DoublyLinkedList:
def __init__(self):
self.head = None

28. 21
3.2.3 Circular Linked List
In a circular linked list, the last node points back to the first node, forming a circle. It
can be singly or doubly linked.
(Using Singly Circular)
class Node:
def __init__(self, data):
self.data = data
self.next = None
class CircularLinkedList:
def __init__(self):
self.head = None
3.2.4 Memory Usage
Singly Linked List: Requires memory for the data and a pointer to the next
node.
Doubly Linked List: Requires memory for the data, a pointer to the next
node, and a pointer to the previous node.
Circular Linked List: Similar to singly or doubly linked lists in memory
usage, but with the last node pointing back to the first node.
3.3 Stack
A stack is a linear data structure that follows a particular order for operations. The order
may be LIFO (Last In First Out) or FILO (First In Last Out).
class Stack:
def __init__(self):
self.items = []
def push(self, item):
self.items.append(item)
def pop(self):
if not self.is_empty():

29. 22
return self.items.pop()
def peek(self):
if not self.is_empty():
return self.items[-1]
def is_empty(self):
return len(self.items) == 0
def size(self):
return len(self.items)
# Example usage
stack = Stack()
stack.push(1)
stack.push(2)
stack.push(3)
print(stack.peek()) # Output: 3
print(stack.pop()) # Output: 3
print(stack.pop()) # Output: 2
print(stack.size()) # Output: 1
3.3.1 LIFO Principle
The LIFO (Last In, First Out) principle means that the last element added to the
stack will be the first one to be removed. Think of it like a stack of plates: the last
plate placed on top is the first one to be taken off.
3.3.2 Common Operations
Push:
Adds an element to the top of the stack.
Time Complexity: O(1) (since adding an element to the end of the list is
a constant-time operation).

30. 23
Pop:
Removes and returns the top element of the stack.
Time Complexity: O(1) (since removing an element from the end of the
list is a constant-time operation).
Peek:
Returns the top element of the stack without removing it.
Time Complexity: O(1) (since accessing the last element of the list is a
constant-time operation).
3.3.3 Memory Usage
Memory usage for a stack is generally O(n), where n is the number of elements in
the stack. This is because we need to store each element in memory.
3.4 Queue
A queue is a linear data structure that follows the First In, First Out (FIFO) principle. This
means that the first element added to the queue will be the first one to be removed.
Simple Queue
Circular Queue
Priority Queue
3.4.1 FIFO Principle
The FIFO (First In, First Out) principle is analogous to a line of people at a ticket
counter. The first person to join the line is the first one to be served and leave the
line.
3.4.2 Simple Queue
A simple queue operates on the FIFO principle. Elements are added at the rear and
removed from the front.
class SimpleQueue:
def __init__(self):
self.queue = []

31. 24
def enqueue(self, item):
self.queue.append(item)
def dequeue(self):
if not self.is_empty():
return self.queue.pop(0)
def is_empty(self):
return len(self.queue) == 0
def front(self):
if not self.is_empty():
return self.queue[0]
# Example usage
queue = SimpleQueue()
queue.enqueue(1)
queue.enqueue(2)
queue.enqueue(3)
print(queue.front()) # Output: 1
print(queue.dequeue()) # Output: 1
print(queue.dequeue()) # Output: 2
print(queue.is_empty()) # Output: False
3.4.3 Circular Queue
In a circular queue, the last position is connected back to the first position to form a
circle. This improves the utilization of space.
class CircularQueue:
def __init__(self, size):
self.queue = [None] * size
self.max_size = size
self.front = -1

32. 25
self.rear = -1
def enqueue(self, item):
if ((self.rear + 1) % self.max_size ==
self.front):
print("Queue is full")
elif self.front == -1:
self.front = 0
self.rear = 0
self.queue[self.rear] = item
else:
self.rear = (self.rear + 1) % self.max_size
self.queue[self.rear] = item
def dequeue(self):
if self.front == -1:
print("Queue is empty")
elif self.front == self.rear:
temp = self.queue[self.front]
self.front = -1
self.rear = -1
return temp
else:
temp = self.queue[self.front]
self.front = (self.front + 1) % self.max_size
return temp
def is_empty(self):
return self.front == -1
def front(self):
if self.is_empty():
return None

33. 26
return self.queue[self.front]
# Example usage
circular_queue = CircularQueue(5)
circular_queue.enqueue(1)
circular_queue.enqueue(2)
circular_queue.enqueue(3)
print(circular_queue.front()) # Output: 1
print(circular_queue.dequeue()) # Output: 1
print(circular_queue.dequeue()) # Output: 2
print(circular_queue.is_empty()) # Output: False
3.4.4 Priority Queue
In a priority queue, elements are removed based on their priority, not just their
position in the queue. Higher priority elements are dequeued before lower priority
elements.
import heapq
class PriorityQueue:
def __init__(self):
self.queue = []
def enqueue(self, item, priority):
heapq.heappush(self.queue, (priority, item))
def dequeue(self):
if not self.is_empty():
return heapq.heappop(self.queue)[1]
def is_empty(self):
return len(self.queue) == 0
def front(self):

34. 27
if not self.is_empty():
return self.queue[0][1]
# Example usage
priority_queue = PriorityQueue()
priority_queue.enqueue('A', 2)
priority_queue.enqueue('B', 1)
priority_queue.enqueue('C', 3)
print(priority_queue.front()) # Output: B
print(priority_queue.dequeue()) # Output: B
print(priority_queue.dequeue()) # Output: A
print(priority_queue.is_empty()) # Output: False
3.5 Hash
A hash table is a data structure that maps keys to values. It uses a hash function to compute
an index into an array of buckets or slots, from which the desired value can be found. Hash
tables are highly efficient for lookups, insertions, and deletions.
def simple_hash(key, size):
return hash(key) % size
size = 10
index = simple_hash("example", size)
print(index) # This will print the index for the key
"example"
3.5.1 Hashing Mechanisms
Hashing is the process of converting a given key into a unique index. The hash
function processes the input and returns a fixed-size string or number that typically
serves as an index in the array.

35. 28
3.5.2 Collision Resolution Techniques
Since multiple keys can hash to the same index (a situation known as a collision), we
need ways to handle these collisions. The two primary techniques are chaining and
open addressing.
Chaining: In chaining, each bucket contains a linked list (or another
secondary data structure) of all elements that hash to the same index. This
way, multiple values can be stored at each index.
Open Addressing: In open addressing, when a collision occurs, the
algorithm searches for the next available slot within the table itself. There
are several strategies for open addressing, including linear probing,
quadratic probing, and double hashing.
3.6 Tree
A tree is a non-linear hierarchical data structure that consists of nodes connected by edges.
Each node contains a value, and nodes are organized in a hierarchical manner. The topmost
node is called the root, and each node has zero or more child nodes.
Binary Tree
Binary Search Tree (BST)
3.6.1 Binary Tree
A binary tree is a type of tree where each node has at most two children, referred to
as the left child and the right child.
class Node:
def __init__(self, data):
self.data = data
self.left = None
self.right = None
class BinaryTree:
def __init__(self):
self.root = None

36. 29
3.6.2 Binary Search Tree (BST)
A binary search tree (BST) is a binary tree with the additional property that for any
given node, the value of the left child is less than the value of the node, and the value
of the right child is greater than the value of the node. This property makes searching
operations efficient.
class Node:
def __init__(self, data):
self.data = data
self.left = None
self.right = None
class BinarySearchTree:
def __init__(self):
self.root = None
def insert(self, data):
if self.root is None:
self.root = Node(data)
else:
self._insert(self.root, data)
def _insert(self, current_node, data):
if data < current_node.data:
if current_node.left is None:
current_node.left = Node(data)
else:
self._insert(current_node.left, data)
elif data > current_node.data:
if current_node.right is None:
current_node.right = Node(data)
else:
self._insert(current_node.right, data)

37. 30
3.6.3 Traversal Method
Tree traversal methods are used to visit all the nodes in a tree and perform an
operation (e.g., printing the node's value) on each node. The three common traversal
methods for binary trees are in-order, pre-order, and post-order.
In-order Traversal:
Traverse the left subtree
Visit the root node
Traverse the right subtree
Pre-order Traversal:
Visit the root node
Traverse the left subtree
Traverse the right subtree
Post-order Traversal:
Traverse the left subtree
Traverse the right subtree
Visit the root node
3.7 Graph
A graph is a data structure that consists of a finite set of vertices (or nodes) and a set of
edges connecting them. Graphs are used to model pairwise relations between objects.
Directed Graph
Undirected Graph
3.8.1 Directed Graph (Digraph)
In a directed graph, edges have a direction. This means that each edge is an ordered
pair of vertices, representing a one-way relationship.
A → B → C
In this example, you can go from A to B and from B to C, but not the other way
around.

38. 31
3.7.2 Undirected Graph
In an undirected graph, edges do not have a direction. Each edge is an unordered
pair of vertices, representing a two-way relationship.
A - B – C
In this example, you can travel between A and B or B and C in either direction.
3.8.3 Representation
Graphs can be represented in several ways, with the most common being the
adjacency matrix and adjacency list.
Adjancency Matrix: An adjacency matrix is a 2D array of size V x V
where V is the number of vertices. If there is an edge from vertex i to vertex
j, then the matrix at position (i, j) will be 1 (or the weight of the edge if it's
weighted). Otherwise, it will be 0.
Adjacency List: An adjacency list is an array of lists. The array size is equal
to the number of vertices. Each entry i in the array contains a list of vertices
to which vertex i is connected.
3.8 Heap
A heap is a specialized tree-based data structure that satisfies the heap property. In a heap,
for every node i, the value of i is either greater than or equal to (in a max-heap) or less than
or equal to (in a min-heap) the value of its children, if they exist.
3.9.1 Min-Heap vs Max-Heap
Min-Heap: In a min-heap, the value of the parent node is always less than or
equal to the values of its children. The root node has the smallest value.
1
/ \
3 2
/ \
4 5

39. 32
Max-Heap: In a max-heap, the value of the parent node is always greater
than or equal to the values of its children. The root node has the largest value
5
/ \
3 4
/ \
1 2
3.8.2 Common Operations
Insert: Inserting an element into a heap involves adding the new element to
the end of the heap and then restoring the heap property by comparing the
new element with its parent and swapping if necessary. This process is called
"heapifying up".
class MinHeap:
def __init__(self):
self.heap = []
def insert(self, key):
self.heap.append(key)
self._heapify_up(len(self.heap) - 1)
def _heapify_up(self, index):
parent_index = (index - 1) // 2
if index > 0 and self.heap[index] <
self.heap[parent_index]:
self.heap[index],
self.heap[parent_index] = self.heap[parent_index],
self.heap[index]
self._heapify_up(parent_index)
# Example usage
min_heap = MinHeap()
min_heap.insert(3)

40. 33
min_heap.insert(1)
min_heap.insert(6)
min_heap.insert(5)
print(min_heap.heap) # Output: [1, 3, 6, 5]
Delete (Extract Min/Max): Deleting the root element from a heap involves
removing the root and replacing it with the last element in the heap. The heap
property is then restored by comparing the new root with its children and
swapping if necessary. This process is called "heapifying down".
(Max-Heap)
def delete_min(self):
if len(self.heap) == 0:
return None
if len(self.heap) == 1:
return self.heap.pop()
root = self.heap[0]
self.heap[0] = self.heap.pop()
self._heapify_down(0)
return root
def _heapify_down(self, index):
smallest = index
left_child = 2 * index + 1
right_child = 2 * index + 2
if left_child < len(self.heap) and
self.heap[left_child] < self.heap[smallest]:
smallest = left_child
if right_child < len(self.heap) and
self.heap[right_child] < self.heap[smallest]:
smallest = right_child
if smallest != index:

41. 34
self.heap[index], self.heap[smallest] =
self.heap[smallest], self.heap[index]
self._heapify_down(smallest)
Peek: Peeking at the heap returns the root element without removing it. In a
min-heap, this is the smallest element; in a max-heap, this is the largest
element.
def peek(self):
if len(self.heap) > 0:
return self.heap[0]
return None

42. 35
Chapter – 4
ALGORITHMS
4.1 Introduction
An algorithm is a step-by-step procedure or formula for solving a problem. They are
essential building blocks in computer science and are used to automate tasks and solve
complex problems efficiently.
Search Algorithms
Divide and Conquer
Sorting Algorithms
Greedy Algorithms
Backtracking
Graph Algorithms
Bit Manipulation
4.2 Time and Space Complexity
Time and space complexity are metrics used to analyze the efficiency of an algorithm. They
help us understand how the algorithm performs as the input size grows.
Time Complexity: This measures the amount of time an algorithm takes to
complete as a function of the size of the input. It's usually expressed using Big O
notation (e.g., O(n), O(log n)), which gives an upper bound on the growth rate.
Space Complexity: This measures the amount of memory an algorithm uses in
relation to the size of the input. Like time complexity, it's also expressed using Big O
notation.
4.2.1 Why is it Necessary?
Understanding time and space complexity is crucial for several reasons:
Efficiency: Helps in choosing the most efficient algorithm for a given
problem.
Scalability: Ensures that the algorithm can handle large inputs.

43. 36
Optimization: Helps in identifying potential bottlenecks and optimizing the
code.
Resource Management: Ensures efficient use of computational resources.
4.2.3 How to Calculate Time and Space Complexity?
Let's consider an example in Python to illustrate this.
Calculating the Sum of a List:
def sum_of_list(nums):
total = 0 # O(1) space
for num in nums: # Loop runs 'n' times, where 'n' is
the length of the list
total += num # O(1) space for the addition
return total # O(1) time
# Example usage
numbers = [1, 2, 3, 4, 5]
result = sum_of_list(numbers)
print(result)
Time Complexity Analysis:
Initialization: total = 0 takes constant time, O(1).
Loop: The loop runs 'n' times, where 'n' is the length of the list, so this is
O(n).
Addition: Adding each element takes constant time, O(1).
Overall, the time complexity is O(n).
Space Complexity Analysis:
Variables: total uses O(1) space.
Input list: The input list nums itself takes O(n) space.
Overall, the space complexity is O(n).

44. 37
4.3 Search Algorithms
Used to search for an element within a data structure.
Linear Search
Binary Search
Depth-First Search (DFS)
Breadth-First Search (BFS)
4.3.1 Linear Search
Linear search is the simplest search algorithm. It checks every element in the list
sequentially until the target value is found or the list ends.
def linear_search(arr, target):
for i in range(len(arr)):
if arr[i] == target:
return i # Return the index of the target
return -1 # Return -1 if the target is not found
# Example usage
arr = [3, 5, 1, 4, 2]
target = 4
print(linear_search(arr, target)) # Output: 3
4.3.2 Binary Search
Binary search is an efficient algorithm for finding an item from a sorted list of
items. It works by repeatedly dividing the search interval in half.
def binary_search(arr, target):
left, right = 0, len(arr) - 1
while left <= right:
mid = (left + right) // 2
if arr[mid] == target:
return mid # Return the index of the target

45. 38
elif arr[mid] < target:
left = mid + 1
else:
right = mid - 1
return -1 # Return -1 if the target is not found
# Example usage
arr = [1, 2, 3, 4, 5]
target = 3
print(binary_search(arr, target)) # Output: 2
4.3.3 Depth-First Search (DFS)
Depth-first search (DFS) is a graph traversal algorithm that starts at the root node
and explores as far as possible along each branch before backtracking.
def dfs(graph, start, visited=None):
if visited is None:
visited = set()
visited.add(start)
print(start, end=' ')
for neighbor in graph[start]:
if neighbor not in visited:
dfs(graph, neighbor, visited)
# Example usage
graph = {
'A': ['B', 'C'],
'B': ['D', 'E'],
'C': ['F'],

46. 39
'D': [],
'E': ['F'],
'F': []
}
dfs(graph, 'A') # Output: A B D E F C
4.3.4 Breadth-First Search (BFS)
Breadth-first search (BFS) is a graph traversal algorithm that starts at the root node
and explores all neighbors at the present depth before moving on to nodes at the next
depth level.
from collections import deque
def bfs(graph, start):
visited = set()
queue = deque([start])
visited.add(start)
while queue:
vertex = queue.popleft()
print(vertex, end=' ')
for neighbor in graph[vertex]:
if neighbor not in visited:
visited.add(neighbor)
queue.append(neighbor)
# Example usage
graph = {
'A': ['B', 'C'],
'B': ['D', 'E'],
'C': ['F'],
'D': [],

47. 40
'E': ['F'],
'F': []
}
bfs(graph, 'A') # Output: A B C D E F
4.4 Divide and Conquer
Divide and Conquer is an algorithmic paradigm that involves breaking a problem into
smaller subproblems, solving each subproblem independently, and then combining their
solutions to solve the original problem. This approach is particularly effective for problems
that can be recursively divided into similar subproblems.
Merge Sort
Quick Sort
Binary Search
4.4.1 Steps in Divide and Conquer
Divide: Break the problem into smaller subproblems of the same type.
Conquer: Solve the subproblems recursively.
Combine: Merge the solutions of the subproblems to get the solution to the
original problem.
4.5 Sorting Algorithms
Used to arrange data in a particular order.
Bubble Sort
Merge Sort
Quick Sort
4.5.1 Bubble Sort
Bubble Sort is a simple comparison-based sorting algorithm. It works by repeatedly
stepping through the list, comparing adjacent elements, and swapping them if they
are in the wrong order. This process is repeated until the list is sorted.

48. 41
def bubble_sort(arr):
n = len(arr)
for i in range(n):
for j in range(0, n-i-1):
if arr[j] > arr[j+1]:
arr[j], arr[j+1] = arr[j+1], arr[j]
return arr
# Example usage
arr = [64, 34, 25, 12, 22, 11, 90]
print(bubble_sort(arr)) # Output: [11, 12, 22, 25, 34,
64, 90]
Time Complexity:
Worst and Average Case: O(n
2
)
Best Case: O(n) (when the array is already sorted)
4.5.2 Merge Sort
Merge Sort is a divide and conquer algorithm. It divides the input array into two
halves, calls itself for the two halves, and then merges the two sorted halves.
def merge_sort(arr):
if len(arr) > 1:
mid = len(arr) // 2
left_half = arr[:mid]
right_half = arr[mid:]
merge_sort(left_half)
merge_sort(right_half)
i = j = k = 0
while i < len(left_half) and j < len(right_half):
if left_half[i] < right_half[j]:

49. 42
arr[k] = left_half[i]
i += 1
else:
arr[k] = right_half[j]
j += 1
k += 1
while i < len(left_half):
arr[k] = left_half[i]
i += 1
k += 1
while j < len(right_half):
arr[k] = right_half[j]
j += 1
k += 1
return arr
# Example usage
arr = [38, 27, 43, 3, 9, 82, 10]
print(merge_sort(arr)) # Output: [3, 9, 10, 27, 38, 43,
82]
Time Complexity:
Worst, Average, and Best Case: O(n log n)
4.5.3 Quick Sort
Quick Sort is a divide and conquer algorithm. It selects a 'pivot' element and
partitions the array around the pivot, such that elements on the left of the pivot are
less than the pivot and elements on the right are greater. The process is then
recursively applied to the sub-arrays. This leads to an efficient sorting process with
an average time complexity of O(n log n). Additionally, Quick Sort is often faster in
practice due to its efficient cache performance and reduced overhead from fewer
memory writes compared to other sorting algorithms like Merge Sort.

50. 43
def partition(arr, low, high):
pivot = arr[high]
i = low - 1
for j in range(low, high):
if arr[j] <= pivot:
i += 1
arr[i], arr[j] = arr[j], arr[i]
arr[i + 1], arr[high] = arr[high], arr[i + 1]
return i + 1
def quick_sort(arr, low, high):
if low < high:
pi = partition(arr, low, high)
quick_sort(arr, low, pi - 1)
quick_sort(arr, pi + 1, high)
return arr
# Example usage
arr = [10, 7, 8, 9, 1, 5]
print(quick_sort(arr, 0, len(arr) - 1)) # Output: [1,
5, 7, 8, 9, 10]
Time Complexity:
Worst Case: O(n
2
) (when the pivot is the smallest or largest element)
Average and Best Case: O(n log n)
4.6 Greedy Algorithms
Greedy algorithms are a type of algorithmic paradigm that makes a series of choices by
selecting the best option available at each step. The goal is to find an overall optimal
solution by making a locally optimal choice at each stage.

51. 44
4.6.1 Prim’s Algorithm
Prim's algorithm is a greedy algorithm that finds the Minimum Spanning Tree
(MST) for a weighted undirected graph. The MST is a subset of the edges that
connects all vertices in the graph with the minimum total edge weight and without
any cycles.
import heapq
def prims_algorithm(graph, start):
mst = []
visited = set()
min_heap = [(0, start)] # (weight, vertex)
while min_heap:
weight, current_vertex = heapq.heappop(min_heap)
if current_vertex in visited:
continue
visited.add(current_vertex)
mst.append((weight, current_vertex))
for neighbor, edge_weight in
graph[current_vertex]:
if neighbor not in visited:
heapq.heappush(min_heap, (edge_weight,
neighbor))
return mst
# Example usage
graph = {
'A': [('B', 1), ('C', 3)],
'B': [('A', 1), ('C', 7), ('D', 5)],
'C': [('A', 3), ('B', 7), ('D', 12)],
'D': [('B', 5), ('C', 12)]

52. 45
}
mst = prims_algorithm(graph, 'A')
print(mst) # Output: [(0, 'A'), (1, 'B'), (5, 'D'), (3,
'C')]
Steps of Prim’s Algorithm:
Initialize a tree with a single vertex, chosen arbitrarily from the graph.
Grow the tree by one edge: choose the minimum weight edge from the graph
that connects a vertex in the tree to a vertex outside the tree.
Repeat step 2 until all vertices are included in the tree.
4.7 Backtracking
Backtracking is an algorithmic paradigm that tries to build a solution incrementally, one
piece at a time. It removes solutions that fail to meet the conditions of the problem at any
point in time (called constraints) as soon as it finds them. Backtracking is useful for solving
constraint satisfaction problems, where you need to find an arrangement or combination
that meets specific criteria.
N-Queen’s Algorithm
4.7.1 N-Queen’s Algorithm
The N-Queens problem involves placing N queens on an N×N chessboard so that no
two queens threaten each other. This means no two queens can share the same row,
column, or diagonal. This classic problem is a perfect example of a backtracking
algorithm. Solutions are found by placing queens one by one in different columns,
starting from the leftmost column, and backtracking when a conflict is detected.
def is_safe(board, row, col, N):
for i in range(col):
if board[row][i] == 1:
return False
for i, j in zip(range(row, -1, -1), range(col, -1,
-1)):
if board[i][j] == 1:

53. 46
return False
for i, j in zip(range(row, N, 1), range(col, -1,
-1)):
if board[i][j] == 1:
return False
return True
def solve_n_queens_util(board, col, N):
if col >= N:
return True
for i in range(N):
if is_safe(board, i, col, N):
board[i][col] = 1
if solve_n_queens_util(board, col + 1, N):
return True
board[i][col] = 0 # Backtrack
return False
def solve_n_queens(N):
board = [[0] * N for _ in range(N)]
if not solve_n_queens_util(board, 0, N):
return "Solution does not exist"
return board
# Example usage
N = 4
solution = solve_n_queens(N)
for row in solution:
print(row)

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Output:
[0, 0, 1, 0]
[1, 0, 0, 0]
[0, 0, 0, 1]
[0, 1, 0, 0]
4.8 Graph Algorithm
Graph algorithms are used to solve problems related to graph theory, which involves
vertices (or nodes) and edges connecting them. These algorithms help in understanding and
manipulating the structure and properties of graphs. Here are some of the most important
graph algorithms:
Dijkstra’s Algorithm
4.8.1 Dijkstra’s Algorithm
Dijkstra's algorithm is used to find the shortest path from a source vertex to all
other vertices in a graph with non-negative edge weights. It uses a priority queue to
repeatedly select the vertex with the smallest distance, updates the distances of its
neighbors, and continues until all vertices have been processed.
import heapq
def dijkstra(graph, start):
distances = {vertex: float('infinity') for vertex in
graph}
distances[start] = 0
priority_queue = [(0, start)]
while priority_queue:
current_distance, current_vertex =
heapq.heappop(priority_queue)
if current_distance > distances[current_vertex]:
continue
for neighbor, weight in graph[current_vertex]:

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distance = current_distance + weight
if distance < distances[neighbor]:
distances[neighbor] = distance
heapq.heappush(priority_queue, (distance,
neighbor))
return distances
# Example usage
graph = {
'A': [('B', 1), ('C', 4)],
'B': [('A', 1), ('C', 2), ('D', 5)],
'C': [('A', 4), ('B', 2), ('D', 1)],
'D': [('B', 5), ('C', 1)]
}
start_vertex = 'A'
print(dijkstra(graph, start_vertex)) # Output: {'A': 0,
'B': 1, 'C': 3, 'D': 4}
4.8.2 Steps of Dijkstra’s Algorithm
Initialise: Set source vertex distance to 0; all other vertices to infinity. Add
source vertex to the priority queue with distance 0.
Relaxation: Extract vertex with minimum distance. For each neighbor,
update distances if a shorter path is found, and add neighbor to priority
queue.
Repeat: Keep extracting the minimum distance vertex and updating
neighbors until the priority queue is empty.

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CONCLUSION
In this enriching exploration of data structures and algorithms, a profound understanding
has been unearthed, weaving a tapestry of computational elegance. The journey
commenced with the foundations of linked lists, where the fluid interplay of singly, doubly,
and circular variations showcased the graceful management of dynamic data. Through the
orchestration of stacks and queues, the principles of LIFO and FIFO came to life,
demonstrating the seamless handling of operations in a structured manner. Hash tables
revealed the magic of hashing, where keys and values found their place through efficient
collision resolution techniques. The branching paths of binary trees and the intricate
networks of graphs, traversed through the methodologies of DFS and BFS, painted a vivid
picture of hierarchical and interconnected data structures.
As the narrative unfolded, the power of algorithms took center stage. Sorting algorithms
like bubble, merge, and quick sort choreographed the transformation of unordered data into
organized sequences, each with its distinct rhythm and efficiency. Greedy algorithms and
backtracking illuminated the path to optimization and constraint satisfaction, with Prim's
algorithm and the N-Queens problem exemplifying their prowess. The divide and conquer
approach brought forth the beauty of breaking problems into manageable fragments,
elegantly demonstrated by merge sort and binary search. The climax was marked by the
elegance of Dijkstra's algorithm, a masterful technique that navigates the vertices of a graph
to unveil the shortest paths with precision and clarity. This journey, rich with insights and
computational artistry, stands as a testament to the profound depth and beauty of data
structures and algorithms in the realm of computer science.

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REFRENCES
[1.] Carnes, B. (2021) Learn algorithms and data structures in Python.
https://www.freecodecamp.org/news/learn-algorithms-and-data-structures-in-
python/.
[2.] DSA using Python (no date). https://premium.mysirg.com/learn/DSA-using-
Python.
[3.] GeeksforGeeks (2024) Learn DSA with Python | Python Data Structures
and Algorithms. https://www.geeksforgeeks.org/python-data-structures-and-
algorithms/.
[4.] House of the Chronicles (no date).
https://www.playbook.com/s/ayushbhattacharya/be4sABV1fU9xKKJ657f8okEr.
[5.] Larson, Q. (2020) Python Data Science – a free 12-Hour course for
beginners. Learn Pandas, NUMPy, Matplotlib, and more.
https://www.freecodecamp.org/news/python-data-science-course-matplotlib-
pandas-numpy/.
[6.] Learn Data Structures and Algorithms with Python | Codecademy (no date).
https://www.codecademy.com/learn/learn-data-structures-and-algorithms-with-
python.
[7.] Programming, data Structures and Algorithms using Python - course (no
date). https://onlinecourses.nptel.ac.in/noc25_cs59/preview.