VARUVAN VADIVELAN INSTITUTE OF TECHNOLOGY

DHARMAPURI
DEPARTMENT OF B-TECH AI&DS
Subject code & Name: AD3301 Design and Analysis of Algorithm
INTERNAL TEST -1
PART – A (2*5=10)
1. Define algorithm with diagram
2. what is recurrence equation?
3. Write an algorithm using Recursive function to fine Sum of n
numbers?
4. What is Time & Space Complexity?
5. Compare the order of growth in n(n-)/2 and n2

PART – B (10*5 = 50)

1. Write short notes on algorithm visualization?

2. Discuss various methods used for mathematical analysis of recursive
algorithm.
3. State the general plan for analyzing the time efficiency of non-recursive
algorithms and explain with an example.
4. Explain asymptotic notations in detail
5. Discuss in detail about fundamentals of algorithm problem solving.
 VARUVAN VADIVELAN INSTITUTE OF TECHNOLOGY
DHARMAPURI
DEPARTMENT OF B-TECH AI&DS
Subject code & Name: AD3301 Design and Analysis of Algorithm
INTERNAL TEST -1 ANSWER KEY
PART – A (2*5=10)

1. Define algorithm with diagram

An algorithm is a sequence of unambiguous instructions for solving
problem

Input output

2. what is recurrence equation?

A recurrence equation, also known as a recurrence relation or a
recurrence relation equation, is a mathematical equation or formula that expresses
a sequence or series of values in terms of one or more previous terms in the same
sequence. In other words, it describes how a value at a specific position in the
sequence depends on values at earlier positions.
T(n) = T(n-1)+n from >0
T(0) = 0 - initial condition

3. Write an algorithm using Recursive function to fine Sum of n

numbers?
 Algorithm: SumOfNRecursive
Input:
n - a non-negative integer
Output:
sum - the sum of the first 'n' natural numbers
Function SumOfNRecursive(n):
// Base case: If n is 0, the sum is 0.
if n == 0:
return 0
// Recursive case: Sum of first 'n' natural numbers is n + Sum of first 'n-1' natural numbers.
else:
return n + SumOfNRecursive(n - 1)
// Example usage:
n=5
result = SumOfNRecursive(n)
Display "The sum of the first", n, "natural numbers is", result

4. What is Time & Space Complexity?

1. Time Complexity:
 Time complexity measures how the running time of an algorithm grows

with the size of the input. It provides an estimate of the number of basic
operations an algorithm will perform as a function of the input size.
 Time complexity is typically expressed using big O notation (e.g., O(n),

O(log n), O(n^2), etc.), which describes an upper bound on the growth rate
of the running time.
 Algorithms with lower time complexity are more efficient and desirable

because they can handle larger inputs in less time.

2. Space Complexity:
 Space complexity measures the amount of memory space (in terms of

auxiliary space, i.e., extra space other than the input) an algorithm requires
to solve a problem as a function of the input size.
  Space complexity is also expressed using big O notation, similar to time
complexity.
 Algorithms with lower space complexity are more memory-efficient,
which is crucial when dealing with limited memory resources, such as in
embedded systems or when working with large datasets.


5. Compare the order of growth in n(n-)/2 and n2

1. n(n-1)/2:
 This expression represents a time complexity of O(n^2).
 It arises from algorithms or situations where you have a nested loop, and
the inner loop depends on the outer loop's variable.
 The (n-1)/2 part doesn't significantly change the order of growth because,
in big O notation, we focus on the most significant term, which is n^2 in
this case. So, it's still considered quadratic time complexity.
2. n^2:
 This expression represents a time complexity of O(n^2).
 It indicates a quadratic growth rate.
 Algorithms with a straightforward double-nested loop structure often
result in a time complexity of O(n^2).

PART B
1. Write short notes on algorithm visualization?
Algorithm visualization is a method of representing algorithms visually to aid in understanding
their behaviour, logic, and execution processes. It can be an invaluable tool for both teaching and learning
algorithms, as well as for debugging and analysing the efficiency of algorithms. Here are some short notes on
algorithm visualization:

1. Enhancing Understanding: Visualization makes complex algorithms more accessible by providing a

graphical or interactive representation of their inner workings. It helps students and developers grasp the
logic and flow of algorithms more intuitively.
2. Step-by-Step Representation: Algorithm visualization often shows how an algorithm progresses step by
step, making it easier to follow its execution. This can be particularly helpful in educational settings.
3. Debugging and Testing: Visualizing algorithms can aid in identifying and resolving errors or
inefficiencies in the code. When you can see the algorithm's behavior, it becomes easier to pinpoint issues.
4. Time and Space Complexity Analysis: Visualization tools can help in analyzing the time and space
complexity of algorithms by tracking and displaying the number of operations and memory usage at each
step.
 5. Interactive Learning: Many algorithm visualization tools allow for user interaction, enabling learners to
experiment with different inputs and see how algorithms respond. This hands-on experience can deepen
understanding.
6. Variety of Representations: Algorithms can be visualized in various ways, including flowcharts,
pseudocode, animations, and interactive web-based tools. The choice of representation depends on the
target audience and the specific algorithm being visualized.
7. Common Algorithms: Visualization is frequently used to explain fundamental algorithms, such as sorting
algorithms (e.g., bubble sort, quicksort), searching algorithms (e.g., binary search), and graph algorithms
(e.g., Dijkstra's algorithm).
8. Educational Tools: Algorithm visualization is often used in educational environments, such as computer
science courses, to teach algorithmic concepts. It helps students gain a deeper understanding of theoretical
concepts and practical implementation.
9. Online Resources: There are various online platforms and tools that provide algorithm visualization
resources and interactive tutorials, making it easier for individuals to learn and explore algorithms.
10. Research and Development: Algorithm visualization is not limited to education; it is also used in research
and software development to understand and optimize algorithms and data structures.

In conclusion, algorithm visualization is a valuable technique for understanding, teaching, and analyzing
algorithms. It simplifies complex concepts, aids in debugging, and enhances the learning experience, making it an
essential tool in the field of computer science and beyond.

2. Discuss various methods used for mathematical analysis of

recursive
algorithm.
Mathematical analysis of recursive algorithms is crucial for understanding their behavior, time complexity, and
efficiency. Several methods are commonly used for such analysis:

1. Recurrence Relations:
 Recurrence relations are equations that describe the relationship between the time complexity of a
recursive algorithm and the size of the input. They help in expressing the time complexity in terms
of smaller subproblems.
 Solving recurrence relations involves finding a closed-form solution using techniques like
substitution, iteration, or master theorem.
2. Master Theorem:
 The master theorem is a powerful tool for analyzing divide-and-conquer algorithms. It provides a
framework for determining the time complexity of recursive algorithms, specifically those that
divide the problem into subproblems of equal size.
 The master theorem helps classify algorithms into different time complexity classes (e.g., O(n),
O(log n), O(n log n), etc.) without going through the full derivation of recurrence relations.
3. Generating Functions:
 Generating functions are a mathematical tool for solving recurrence relations. They provide a way
to represent and manipulate sequences generated by recursive algorithms.
 By transforming the recurrence relation into a generating function and using techniques from
generating function theory, you can derive closed-form solutions.
 4. Asymptotic Notation (Big O, Big Theta, Big Omega):
 Asymptotic notation is used to provide an upper bound (Big O), lower bound (Big Omega), or tight
bound (Big Theta) on the growth rate of a recursive algorithm.
 This method simplifies the analysis by focusing on the most significant terms and constants in the
recurrence relations.
5. Substitution and Iteration Methods:
 Substitution involves assuming a solution and then proving it by induction. You make a guess and
prove its correctness.
 Iteration, on the other hand, involves expanding the recurrence relation iteratively until a pattern or
solution becomes evident.
6. Case Analysis:
 For some algorithms, you may perform a detailed case analysis to determine the time complexity.
This involves considering different scenarios or subproblem sizes and analyzing each case
separately.
7. Amortized Analysis:
 Amortized analysis is used to determine the average time complexity of a sequence of operations in
data structures or algorithms. It provides a more comprehensive view of performance over time.
 Techniques like aggregate analysis, potential method, and accounting method are used in amortized
analysis.
8. Probabilistic Analysis:
 In some cases, when the exact behavior of a recursive algorithm is challenging to determine,
probabilistic analysis can be employed. It involves considering the expected values of time
complexity.
9. Analytical Combinatorics:
 Analytical combinatorics is a branch of mathematics that combines combinatorial and analytic
methods to analyze the performance of algorithms. It's particularly useful for recursive algorithms
that involve combinatorial structures like trees, graphs, and permutations.
10. Recursion Tree Analysis:
 For divide-and-conquer algorithms, creating a recursion tree can be a helpful visual aid in analyzing
their time complexity. The tree illustrates how the problem is divided into subproblems and how
they are solved.

These methods are not mutually exclusive, and the choice of which method to use depends on the specific
characteristics of the recursive algorithm and the level of precision required for the analysis. In practice, a
combination of these methods may be employed to achieve a comprehensive understanding of the algorithm's
performance.

3. State the general plan for analyzing the time efficiency of non-
recursive
algorithms and explain with an example.
1. Determine the Basic Operation:
 Identify the fundamental operation(s) that are executed most frequently in
the algorithm. This is often the operation that dominates the running time.
2. Count the Number of Basic Operations:
 Analyze how many times the basic operation(s) are executed in terms of

the input size. You'll typically express this count as a function of the input
size, denoted as �n.
3. Express the Time Complexity:
 Express the time complexity of the algorithm using Big O notation. This

simplifies the analysis by focusing on the most significant factors affecting

running time.
4. Provide an Example:
 To illustrate this plan, let's consider an example with the following non-

recursive algorithm that finds the maximum element in an array of

integers.

Algorithm FindMax
Input: An array of integers arr of size n.
Output: The maximum element in arr.

max_element = arr[0] # Initialize max_element with the first element

for i from 1 to n-1:
if arr[i] > max_element:
max_element = arr[i] # Update max_element if a larger element is found

Return max_element

 Determine the Basic Operation:

 In this algorithm, the basic operation is the comparison operation

arr[i] > max_element and the assignment operation max_element = arr[i].

 Count the Number of Basic Operations:
 The loop runs from index 1 to n-1, so it performs n-1 iterations.

 In each iteration, we have one comparison and a possible assignment

operation. In the worst case (when the condition arr[i] > max_element is
always true), both operations are executed.
 Express the Time Complexity:
 The time complexity is expressed as �(�)O(n), where �n is


the size of the input array. This means that the running time of the
algorithm grows linearly with the size of the input.
5. Consider Best, Worst, and Average Cases (if applicable):
 In some cases, you might analyze the best-case, worst-case, and average-

case scenarios. In this example, we focused on the worst-case scenario

where the condition is always true for every element in the array. The
best-case scenario is when the first element is the maximum, resulting in a
time complexity of �(1)O(1).
6. Perform Benchmarks and Testing (optional):
 To validate your analysis, you can conduct practical experiments by

measuring the algorithm's performance with various input sizes and

comparing it to the predicted time complexity.

This general plan provides a structured approach to analyze the time efficiency of
non-recursive algorithms, helping you understand how the running time scales with
the input size and make informed decisions about algorithm selection and
optimization.
Regen
3. Explain asymptotic notations in detail
Asymptotic notations are mathematical notations used in computer science and mathematics to describe the
growth rate of functions, particularly when analyzing the time and space complexity of algorithms. They are a way
to express how the performance of an algorithm or function scales as the input size grows to infinity. The three
most commonly used asymptotic notations are Big O, Big Theta, and Big Omega.

1. Big O Notation (O-notation):

 Big O notation represents an upper bound on the growth rate of a function. It provides an
approximation of the worst-case performance.

2. Big Theta Notation (Θ-notation):

 Big Theta notation represents both an upper and lower bound on the growth rate of a function. It
provides a tight bound on the performance.
 3. Big Omega Notation (Ω-notation):
 Big Omega notation represents a lower bound on the growth rate of a function. It provides an
approximation of the best-case performance.

the exact details of implementation or specific hardware. They allow us to make general statements about how the
running time or space requirements of algorithms scale as the input size grows. It's important to choose the
appropriate notation to convey the relevant information about an algorithm's behavior under different scenarios,
such as best, worst, or average cases.

5. Discuss in detail about fundamentals of algorithm problem

solving.
Solving algorithmic problems is a fundamental skill in computer science and programming. To effectively solve
algorithmic problems, you need to follow a structured approach. Here are the fundamentals of algorithm problem
solving:

1. Understanding the Problem:

 The first step in solving an algorithmic problem is to thoroughly understand the problem statement.
Read and re-read the problem description to ensure you have a clear grasp of the problem's
requirements and constraints.
 Identify the input and output requirements, any specific constraints, and any edge cases that need to
be considered.
2. Example Input and Output:
 Work through a few example inputs and outputs to gain a deeper understanding of how the problem
works. This helps in identifying patterns and relationships within the problem.
3. Breaking Down the Problem:
 Decompose the problem into smaller, manageable parts. Divide it into subproblems or components
that are easier to tackle. This is especially important for complex problems.
 Consider what subproblems need to be solved and how they relate to the overall problem.
4. Design an Algorithm:
 Create a high-level plan or algorithm to solve the problem. This plan should outline the major steps
and strategies for solving the problem.
  Determine which data structures and algorithms are most suitable for the task.
5. Pseudocode or Flowchart:
 Write pseudocode or create a flowchart to illustrate your algorithm. Pseudocode is a high-level
description of the solution without using a specific programming language.
 This step helps in translating your plan into a more structured format.
6. Implement the Solution:
 Write the code based on your pseudocode or flowchart. Pay attention to details and consider how
you can efficiently implement your algorithm.
 Use appropriate data structures and programming constructs to achieve your goal.
7. Test the Solution:
 Test your code with a variety of input cases, including normal, boundary, and edge cases. Verify
that your solution produces the correct output and handles all scenarios.
 Debug and fix any issues that arise during testing.
8. Optimize and Refactor:
 After verifying that your solution works correctly, consider optimizing your code. This may involve
improving the time or space complexity of your algorithm or making the code more readable and
efficient.
 Keep in mind that optimization should not compromise code clarity.
9. Analyzing Time and Space Complexity:
 Analyze the time and space complexity of your algorithm to understand its efficiency. Use
asymptotic notations (Big O, Big Theta, Big Omega) to express the growth rates.
 This analysis helps you evaluate the scalability of your solution and compare it to other algorithms.
10. Document Your Solution:
 Document your code with comments, explaining your thought process and any complex parts of the
code. This documentation will be helpful for you and others when revisiting the code.
11. Practice and Learn:
 Problem-solving is a skill that improves with practice. Solve a variety of algorithmic problems to
build your problem-solving skills and explore different techniques.
 Study algorithms and data structures to expand your problem-solving toolkit.
12. Seek Help and Collaborate:
 Don't hesitate to seek help from online resources, forums, or colleagues if you're stuck on a
problem. Collaboration and discussion can provide valuable insights.
 Learning from others' solutions can help you improve your problem-solving abilities.
13. Stay Patient and Persistent:
 Some problems can be challenging, and it may take time to arrive at a solution. Stay patient and
persistent, and don't get discouraged by initial failures.

The fundamentals of algorithm problem solving involve a combination of understanding, planning, coding, testing,
optimizing, and continuous learning. Developing these problem-solving skills is essential for success in computer
science and software development.

Fundamentals of algorithm problem-solving are essential skills for computer scientists, programmers, and anyone
who deals with complex tasks that require systematic and efficient solutions. Here, we'll discuss these
fundamentals in detail:

1. Problem Understanding:
  The first step in algorithm problem-solving is to fully understand the problem. Read the problem
statement carefully, ask questions if necessary, and break it down into its essential components.
 Identify the input, output, constraints, and the problem's context. Make sure you understand the
problem's requirements and constraints.
2. Plan and Strategy:
 Develop a plan and a strategy for solving the problem. Consider various approaches, trade-offs, and
potential algorithms that can be applied.
 Think about the best data structures, control flow, and algorithms for the task. Consider factors like
time and space complexity.
3. Pseudocode and Flowcharts:
 Before diving into coding, create a pseudocode or a flowchart. Pseudocode is a high-level
description of the solution without using a specific programming language.
 This step helps you structure your thoughts, consider the logic, and provides a roadmap for your
code.
4. Algorithm Design:
 Develop a clear algorithm that outlines the major steps to solve the problem. This algorithm serves
as the blueprint for your code.
 Define the sequence of operations, loops, conditionals, and data structures necessary for the
solution.
5. Coding and Implementation:
 Write the code based on your algorithm. Pay attention to details and follow best coding practices.
Write clear and readable code.
 Use appropriate variables, functions, and libraries. Consider code organization and maintainability.
6. Testing and Debugging:
 Test your code with a variety of input cases. Verify that the code produces the correct output for
normal, boundary, and edge cases.
 Debug and fix any issues that arise during testing. Utilize debugging tools and techniques.
7. Optimization:
 After verifying that your solution works correctly, consider optimizing your code. This may involve
improving the time or space complexity of your algorithm.
 Optimize for performance, but not at the expense of code readability and maintainability.
8. Complexity Analysis:
 Analyze the time and space complexity of your algorithm. Use asymptotic notations (Big O, Big
Theta, Big Omega) to express the growth rates.
 This analysis helps you evaluate the scalability of your solution and compare it to other algorithms.
9. Documentation:
 Document your code with comments and explanations. Describe the purpose and logic of complex
sections, and include information about input, output, and the algorithm used.
 Good documentation is invaluable for code maintainability and collaboration.
10. Review and Peer Feedback:
 Have your code reviewed by peers or colleagues. Getting a fresh perspective and feedback can help
identify issues and alternative solutions.
 Embrace a culture of code review to improve your problem-solving skills.
11. Practice and Continuous Learning:
 Problem-solving is a skill that improves with practice. Regularly tackle algorithmic problems to
build and maintain your problem-solving skills.
  Study algorithms and data structures to expand your toolkit and learn from different problem-
solving techniques.
12. Seek Help and Collaborate:
 Don't hesitate to seek help from online resources, forums, or colleagues if you're stuck on a
problem. Collaboration and discussion can provide valuable insights.
 Learning from others' solutions can help you improve your problem-solving abilities.
13. Stay Patient and Persistent:
 Some problems can be challenging, and it may take time to arrive at a solution. Stay patient,
persistent, and maintain a positive attitude when faced with difficulties.

These fundamentals of algorithm problem-solving involve a structured and systematic approach to tackling
complex problems, developing efficient solutions, and continuously improving your problem-solving skills.