FUNDAMENTALS OF
ALGORITHMS
MCS - 2
LECTURE # 1
PREREQUISITES
• Data Structures
• Discrete Mathematics
COURSE RESOURCES
Compulsory Reading Material
• Introduction to Algorithms by
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest & Clifford Stein
(3rd Edition)
Recommended Reading
• Fundamentals of Computer Algorithms by
Eillies Horowitz & Sahni Sartaj (1990)
• Fundamentals of Algorithms by
Gilles Brassard & Paul Bratley (1995)
• Algorithms; Fourth Edition by
Robert Sedgewick & Kevin Wayne.
THE ROLE OF ALGORITHMS IN
COMPUTING
PROBLEM SOLVING
• Programming is a process of problem solving
• Problem solving techniques
• Analyze the problem
• Outline the problem requirements
• Design steps (algorithm) to solve the problem
• Algorithm:
• Step-by-step problem-solving process
• Solution achieved in finite amount of time
PROBLEM SOLVING PROCESS
• Step 1 - Analyze the problem
• Outline the problem and its requirements
• Design steps (algorithm) to solve the problem
• Step 2 - Implement the algorithm
• Implement the algorithm in code
• Verify that the algorithm works
• Step 3 - Maintenance
• Use and modify the program if the problem domain changes
ANALYZE THE PROBLEM
• Thoroughly understand the problem
• Understand problem requirements
• Does program require user interaction?
• Does program manipulate data?
• What is the output?
• If the problem is complex, divide it into subproblems
• Analyze each subproblem as above
DEFINITION
• In simple terms, an algorithm is a series of instructions to solve a problem
(complete a task)
• Problems can be in any form
• Business
• Allocate manpower to maximize profit
• Real Life
• If I go 10 steps forward, I will reach my mailbox.
• I am hungry. How do I order pizza?
• Explain how to tie shoelaces to a five year old child
APPLICATION OF ALGORITHM IN COMP. SCIENCE AREAS
• Compiler construction
• Operating system
• Databases
• AI
• Graphics
• Networking
ALGORITHM & PROGRAMMING
• The idea behind the computer program
• It is a mathematical entity of a program.
• Stays the same independent of
• Which kind of hardware it is running on
• Which programming language it is written in
• Solves a well-specified problem in a general way
• Is specified by
• Describing the set of instances (input) it must work on
• Describing the desired properties of the output
WHAT IS AN ALGORITHM? (CONT’D)
• Before a computer can perform a task, it must have an algorithm that tells it
what to do.
• Informally: “An algorithm is a set of steps that define how a task is
performed.”
• Formally: “An algorithm is an ordered set of unambiguous executable steps,
defining a terminating process.”
• Ordered set of steps: structure!
• Executable steps: doable!
• Unambiguous steps: follow the directions!
• Terminating: must have an end!
GOOD VS. BAD ALGORITHMS
•
All algorithms will have input, perform a process, and produce
output.
A good algorithm should be:
•
•
Simple - relative
•
Complete – account for all inputs & cases
•
Correct (Right)
•
should have appropriate levels of Abstraction. – grouping steps into a single
module
•
Precise
•
Mnemonic - SCRAP
SIMPLICITY
• Simple can be defined as having no unnecessary steps and no
unnecessary complexity. (You may lose points if your algorithm
contains unnecessary steps)
• Each step of a well developed algorithm should carry out one logical
step of the process.
• Avoid something like: “Take 2nd right after you exit at King Street”
IT HAS LEVELS OF ABSTRACTION.
• Example: Add all the scores then divide the sum by the number of students
to get the average.
PRECISION
• Precision means that there is only one way to interpret the
instruction. Unambiguous
• Words like “maybe”, “sometimes” and
business in a well developed algorithm.
“occasionally” have no
• Instead of “maybe”, we can specify the exact circumstances in
which an action will be carried out.
ALGORITHM AS TOOL
• It can be viewed as tool for solving “Computational Problem”
• The algorithm describes a specific Computational Procedure for achieving that
relationship between Input and Output.
• Input<32,22,45,66,12,44> sorting algorithm returns output<12,22,32,44,45,66>
• Input
instance
(Satisfying constraints , compute solution)
• Correct Algorithm provides correct Output
OTHER ALGORITHM ATTRIBUTES
• A good algorithm should be correct.
• A good algorithm should be complete.
• “To be correct, an algorithm must produce results that are correct
and complete given any and all sets of appropriate data.”
• And to be correct, an algorithm must proceed through to a
conclusion.
• Correct
IMPORTANT PROPERTIES OF ALGORITHMS /
(CRITERION FOR ANALYZING ALGORITHMS)
• always returns the desired output for all legal instances
of the problem.
• Unambiguous
• Precise
• Efficient
• Can be measured in terms of
• Running Time
• Memory Space
• Sometimes tends to be more important
PROBLEM SOLVED BY ALGORITHMS
• The Human Genes Project(HGP); sorting info in database,
developing tools for analyzing data).enabling Scientists to
accomplish tasks efficiently.
• Manipulating and managing large volume of data on internet sites
with Clever algorithms.
• Numerical algorithms and number theory is helping E-Commerce to
ensure the privacy of personal info.
• Manufacturing and commercial enterprises use algorithms. For
example politician, oil company, airline.
• Model of Road map(shortest path)
• Mechanical designs
ASSIGNMENT # 1
• Write an algorithm to call your friend.
• Give a real world example that requires Sorting for its solution?
• Write an algorithm for searching ‘Algorithm’s text book’ from Library?
• How does one get to see the film “THREE IDIOTS” in cinema?
GOOD LUCK ! ☻