1. Introduction to Algorithms, Specification of Algorithm, Complexity.pdf

Course Name: Design and Analysis of Algorithm
Topic: Introduction to Algorithms, Specification of Algorithm
Course code : CS3102
Credits : 4
Mode of delivery : Hybrid (PowerPoint presentation)
Faculty : Dr. Ajit Noonia
Email-id : ajit.noonia@jaipur.manipal.edu
B.TECH V SEM CSE
ACADEMIC YEAR: 2024-2025
1

Assessment criteria’s
9/4/2024 2

Course Information
• Course Handout
• Communicate through eMail
• Office hours
• To be communicated
• Grading policy
• Will be communicated as per university guidelines
9/4/2024
CS3102 (DAA), Dept. of CSE
3

Syllabus
Introduction: Algorithm Definition and Criteria of Algorithms, Iterative and Recursive
Algorithms. Performance Analysis: Priori and Posteriori Analysis, Asymptotic Notations,
Space Complexity, Time Complexity, Performance measurement of iterative and recursive
algorithms. Solving Recurrence Relations: Substitution Method, Iterative Method, Recursive
Tree Method, Master Method. Divide and Conquer: Introduction, Binary Search, Finding
Maximum and Minimum, Merge Sort, Quick Sort, Randomized Quick Sort, Integer
Multiplication. Graph Search Algorithm: Graph representation, Breadth First Search, and
Depth First Search. Greedy Strategy: Introduction, Knapsack Problem, Job Sequencing with
Deadlines, Huffman Coding, Union and Find Operation (Set and Disjoint Set), Minimum Cost
Spanning Tree Algorithms (Prim’s and Kruskal’s), Optimal Merge Patterns, Single Source
Shortest Path (Dijkstra’s Algorithm). Dynamic Programming: Introduction, Single Source
Shortest Path (Bellman and Ford Algorithm), All Pair Shortest Path (Floyd Wrashal’s
Algorithm), Optimal Binary Search Trees, 0/1 Knapsack Problem, Travelling Salesperson
Problem, Longest Common Subsequence, Matrix Chain Multiplication, Edit distance.
Backtracking: Introduction, N-Queens Problem, Graph Colouring, and Hamiltonian Cycles.
Branch and Bound: Introduction, FIFO and LC Branch and Bound, 0/1 Knapsack Problem,
Travelling Salesman Problem. String Matching: Naïve String Matching, Rabin Karp Algorithm,
Knuth-Morris-Pratt Algorithm. Complexity Classes: NP, NP-Complete and NP-Hard Problems,
Cook’s Theorem, Polynomial-time reductions, Satisfiability, Reduction from Satisfiability to
Vertex Cover.
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4

More Information
• Textbook
•Introduction to Algorithms 3rd, Cormen,
Leiserson, Rivest and Stein, The MIT Press,
• Fundamentals of Computer Algorithms, 2nd,
Sartaj Sahni, Ellis Horowitz, Sanguthevar
Rajasekaran
• Others
• Introduction to Design & Analysis Computer Algorithm 3rd,
Sara Baase, Allen Van Gelder, Adison-Wesley, 2000.
• Algorithms, Richard Johnsonbaugh, Marcus Schaefer, Prentice
Hall, 2004.
• Introduction to The Design and Analysis of Algorithms 2nd
Edition, Anany Levitin, Adison-Wesley, 2007.
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5

Course Outcomes:
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6
[CS3102.1] Analyze the running times of algorithms using asymptotic analysis. (Bloom’s Level: 4
Analyze).
[CS3102.2] Contrast and design algorithms using the divide-and-conquer and graph-searching
algorithms to solve business problems, hence enhancing analytical skills. (Bloom’s Level: 4 Analyze)
[CS3102.3] Apply the concept of greedy and dynamic programming approaches to solving real-life
problems to enhance entrepreneurship capabilities. (Bloom’s Level: 3 Apply)
[CS3102.4] Utilize the principles of backtracking and branch and bound algorithms to showcase their
functioning and problem-solving capabilities. (Bloom’s Level: 3 Apply)
[CS3102.5] Evaluate various advanced algorithm concepts such as string matching, approximation
algorithms, and complexity classes to enhance employability. (Bloom’s Level: 5 Evaluate)

Goal of the Course
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7
Learning
Learning to solve real
problems that arise
frequently in computer
applications.
Learning
Learning the basic
principles and techniques
used for answering the
question: “How good, or,
how bad is the algorithm”
Getting
Getting to know a group
of “very difficult
problems” categorized as
“NP-Complete”

Introduction: Algorithm Definition
and Criteria of Algorithms,
Iterative and Recursive
Algorithms
9/4/2024 CS3102 (DAA), Dept. of CSE 8

What is an Algorithm?
A finite set of instructions that specifies a sequence of
operations to be carried out to solve a specific problem
or class of problems is called an Algorithm.
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Characteristics of Algorithms
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• It should externally supply zero or more quantities.
Input:
• It results in at least one quantity.
Output:
• Each instruction should be clear.
Definiteness:
• An algorithm should terminate after executing a finite
number of steps.
Finiteness:
• the algorithm must produce the correct output within
a reasonable amount of time and using an acceptable
number of resources (such as memory).
Effectiveness:

Advantages of an Algorithm
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Effective Communication:
Since it is written in a natural language
like English, it becomes easy to
understand the step-by-step solution to
any problem.
Easy Debugging:
A well-designed algorithm facilitates easy
debugging to detect the logical errors
that occurred inside the program.
Easy and Efficient Coding:
An algorithm is nothing but a blueprint of
a program that helps develop a program.
Independent of Programming
Language:
Since it is language-independent, it can
be easily coded by incorporating any
high-level language.

Two Distinct Choices
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How to write an Algorithm
Algorithm Swap(a, b)
{
temp = a;
a = b;
b = temp;
}
9/4/2024 CS3102 (DAA), Dept. of CSE 13

Probably the Oldest Algorithm
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Euclid Algorithm: RecursiveVersion
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Analysis of Algorithm
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Space Complexity:
The space complexity can be
understood as the amount of space
required by an algorithm to run to
completion.
Time Complexity:
Time complexity is a function of input
size n that refers to the amount of time
needed by an algorithm to run to
completion.
The analysis is a process of estimating the efficiency of an algorithm. There
are two fundamental parameters based on which we can analyze the
algorithm:

How to Analyse an Algorithm
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Algorithm Swap(a, b)
{
temp = a;
a = b;
b = temp;
}

Applications
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Frequency Count Method
Algorithm Sum(A, n)
{
s = 0;
for(i=0; i<n; i++)
{
s = s + A[i];
}
return s;
}

Frequency Count Method (Example)
Algorithm Add(A, B, n)
{
for(i=0; i<n; i++)
{
for(j=0; j<n; j++)
{
C[i,j] = A[i,j] + B[i,j]
}
}
}

Frequency Count Method (Example)
Algorithm Multiply(A, B, n)
{
for(i=0; i<n; i++)
{
for(j=0; j<n; j++)
{
C[i,j] = 0;
for(k=0; k<n; k++)
{
C[i,j] = C[i,j] + A[i,k] * B[i,j];
}
}
}
}

Example
for(i=0; i<n; i++)
{
statement;
}

Example
for(i=n; i>0; i--)
{
statement;
}

Example
for(i=0; i<n; i=i+2)
{
statement;
}

Example
for(i=0; i<n; i++)
{
for(j=0; j<n; j++)
{
statement;
}
}

Example
for(i=0; i<n; i++)
{
for(j=0; j<i; j++)
{
statement;
}
}

Example
p=0;
for(i=1; p<=n; i++)
{
p=p+i;
}

Example
for(i=1; i<n; i=i*2)
{
statement;
}

Example
for(i=n; i>=1; i=i/2)
{
statement;
}

Example
for(i=1; i<n; i++)
{
statement;
}
for(j=1; j<n; j++)
{
statement;
}

Example
for(i=0; i<n; i++)
{
for(j=1; j<n; j=j*2)
statement;
}

Observations
1. for(i=0; i<n; i++) O(n)
2. for(i=0; i<n; i+2) n/2 O (n)
3. for(i=n; i>1; i--) O (n)
4. for(i=1; i<n; i=i*2) O(log n) Base 2
5. for(i=1; i<n; i=i*3) O(log n) Base 3
6. for(i=n; i>1; i=i/2) O(log n) Base 2

1. Introduction to Algorithms, Specification of Algorithm, Complexity.pdf

Compare Class of Functions
1<log n < √n < n < n*log n < n2 < n3 < - - - - - - -< 2n < 3n < ----<nn
log n n n^2 2^n
0 1 1 2
1 2 4 4
2
3
4
8
16
64
16
256

1. Introduction to Algorithms, Specification of Algorithm, Complexity.pdf

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