algorithm analysis

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Advanced Algorithms Analysis
By
Dr. Attiqa Rehman
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Objective of This Course
Major objective of this course is:
• Design and analysis of modern algorithms
• Different variants
• Accuracy
• Efficiency
• Comparing efficiencies
• Motivation thinking new algorithms
• Advanced designing techniques
• Real world problems will be taken as examples
• To create feelings about usefulness of this course
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Expected Results
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On successful completion, students will be able to
Argue and prove correctness of algorithms
Derive and solve mathematical models of problems
Reasoning when an algorithm calls certain approach
Analyze average and worst-case running times
Integrating approaches in dynamic and greedy algos.
Use of graph theory in problems solving
Advanced topics such as
• Computational geometry, number theory etc.
Several other algorithms such as
• String matching, NP completeness, approximate
algorithms etc.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Lecture No 1
Introduction
(What, Why and Where Algorithms . . .)
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Today Covered
In this lecture we will cover the following
• What is Algorithm?
• Designing Techniques
• Model of Computation
• Algorithms as a technology
• Algorithms and other technologies
• Importance of algorithms
• Difference in Users and Developers
• Kinds of problems solved by algorithms
• Conclusion
Dr. Attiqa Rehman
Advanced Algorithms Analysis
What is Algorithm?
• A computer algorithm is a detailed step-by-step method
for solving a problem by using a computer.
• An algorithm is a sequence of unambiguous instructions
for solving a problem in a finite amount of time.
• An Algorithm is well defined computational procedure that
takes some value, or set of values, as input and produces
some value, or set of values as output.
• More generally, an Algorithm is any well defined
computational procedure that takes collection of elements
as input and produces a collection of elements as output.
Input
Dr. Attiqa Rehman
Algorithm
output
Advanced Algorithms Analysis
Popular Algorithms, Factors of Dependence
• Most basic and popular algorithms are
– Sorting algorithms
– Searching algorithms
Which algorithm is best?
• Mainly, it depends upon various factors, for
example in case of sorting
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The number of items to be sorted
The extent to which the items are already sorted
Possible restrictions on the item values
The kind of storage device to be used etc.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
One Problem, Many Algorithms
Problem
• The statement of the problem specifies, in general
terms, the desired input/output relationship.
Algorithm
• The algorithm describes a specific computational
procedure for achieving input/output relationship.
Example
• One might need to sort a sequence of numbers
into non-decreasing order.
Algorithms
• Various algorithms e.g. merge sort, quick sort,
heap sorts etc.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Important Designing Techniques
• Brute Force
– Straightforward, naive approach
– Mostly expensive
• Divide-and-Conquer
– Divide into smaller sub-problems
• Iterative Improvement
– Improve one change at a time
• Decrease-and-Conquer
– Decrease instance size
• Transform-and-Conquer
– Modify problem first and then solve it
• Space and Time Tradeoffs
– Use more space now to save time later
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Some of the Important Designing Techniques
• Greedy Approach
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Locally optimal decisions, can not change once made.
Efficient
Easy to implement
The solution is expected to be optimal
Every problem may not have greedy solution
• Dynamic programming
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Decompose into sub-problems like divide and conquer
Sub-problems are dependant
Record results of smaller sub-problems
Re-use it for further occurrence
Mostly reduces complexity exponential to polynomial
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Problem Solving Phases
• Analysis
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How does system work?
Breaking a system down to known components
How components (processes) relate to each other
Breaking a process down to known functions
• Synthesis
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Building tools
Building functions with supporting tools
Composing functions to form a process
How components should be put together?
Final solution
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Problem Solving Process
• Problem
• Strategy
• Algorithm
– Input
– Output
– Steps
• Analysis
– Correctness
– Time & Space
– Optimality
• Implementation
• Verification
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Model of Computation (Assumptions)
• Design assumption
– Level of abstraction which meets our requirements
– Neither more nor less e.g. [0, 1] infinite continuous interval
• Analysis independent of the variations in
– Machine
– Operating system
– Programming languages
– Compiler etc.
• Low-level details will not be considered
• Our model will be an abstraction of a standard
generic single-processor machine, called a random
access machine or RAM.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Model of Computation (Assumptions)
• A RAM is assumed to be an idealized machine
– Infinitely large random-access memory
– Instructions execute sequentially
• Every instruction is in fact a basic operation on two
values in the machines memory which takes unit time.
• These might be characters or integers.
• Example of basic operations include
– Assigning a value to a variable
– Arithmetic operation (+, - , × , /) on integers
– Performing any comparison e.g. a < b
– Boolean operations
– Accessing an element of an array.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Model of Computation (Assumptions)
• In theoretical analysis, computational complexity
– Estimated in asymptotic sense, i.e.
– Estimating for large inputs
• Big O, Omega, Theta etc. notations are used to
compute the complexity
• Asymptotic notations are used because different
implementations of algorithm may differ in efficiency
• Efficiencies of two given algorithm are related
– By a constant multiplicative factor
– Called hidden constant.
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Drawbacks in Model of Computation
First poor assumption
• We assumed that each basic operation takes
constant time, i.e. model allows
– Adding
– Multiplying
– Comparing etc.
two numbers of any length in constant time
• Addition of two numbers takes a unit time!
– Not good because numbers may be arbitrarily
• Addition and multiplication both take unit time!
– Again very bad assumption
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Model of Computation not so Bad
Finally what about Our Model?
• But with all these weaknesses, our model is not so
bad because we have to give the
– Comparison not the absolute analysis of any algorithm.
– We have to deal with large inputs not with the small size
• Model seems to work well describing computational
power of modern nonparallel machines
Can we do Exact Measure of Efficiency ?
• Exact, not asymptotic, measure of efficiency can be
sometimes computed but it usually requires certain
assumptions concerning implementation
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Summary : Computational Model
• Analysis will be performed with respect to this
computational model for comparison of algorithms
• We will give asymptotic analysis not detailed
comparison i.e. for large inputs
• We will use generic uniprocessor random-access
machine (RAM) in analysis
– All memory equally expensive to access
– No concurrent operations
– All reasonable instructions take unit time,
except, of course, function calls
Dr. Attiqa Rehman
Advanced Algorithms Analysis
Conclusion
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What, Why and Where Algorithms?
Designing Techniques
Problem solving Phases and Procedure
Model of computations
– Major assumptions at design and analysis level
– Merits and demerits, justification of assumptions taken
• We proved that algorithm is a technology
• Compared algorithmic technology with others
• Discussed importance of algorithms
– In almost all areas of computer science and engineering
– Algorithms make difference in users and developers
Dr. Attiqa Rehman
Advanced Algorithms Analysis
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