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Questions and Answers
What is a primary reason for writing algorithms in a formal manner?
What is a primary reason for writing algorithms in a formal manner?
Why is it necessary to analyze algorithms despite advancements in computer speed?
Why is it necessary to analyze algorithms despite advancements in computer speed?
What key factor should be considered when analyzing algorithms?
What key factor should be considered when analyzing algorithms?
What advantage does a linked-list-based list implementation offer compared to an array-based list?
What advantage does a linked-list-based list implementation offer compared to an array-based list?
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When should algorithm efficiency typically be considered according to the content?
When should algorithm efficiency typically be considered according to the content?
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What is an essential characteristic of a correct algorithm?
What is an essential characteristic of a correct algorithm?
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According to Al-Khwarizmi’s Golden Principle, what is the first step in solving a complex problem?
According to Al-Khwarizmi’s Golden Principle, what is the first step in solving a complex problem?
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What does the term 'algorism' refer to in its original context?
What does the term 'algorism' refer to in its original context?
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Why is it beneficial to write down an algorithm?
Why is it beneficial to write down an algorithm?
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What was the initial goal of early algorithm studies by mathematicians?
What was the initial goal of early algorithm studies by mathematicians?
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Study Notes
Course Information
- Course: CS341 Algorithms Analysis and Design
- Lecture: 2
- Instructor: Dr. Ahmed Hamza
Algorithm Recap
- Problem Statement: Defines the input/output relationship
- Algorithm: A procedure achieving the relationship
- Definition: A sequence of steps to transform input into output
- Instance: The input needed for computing the solution
- Correct Algorithm: For every input, it halts with the correct output
Brief History of Algorithms
- The study began with mathematicians seeking a general algorithm for each problem type
- Named after Abu Abdullah Muhammad ibn Musa al-Khwarizmi, a 9th-century Persian mathematician
- Dar al-Hikma (House of Wisdom) translated and published scientific and philosophical works, including Greek texts
Al-Khwarizmi's Golden Principles
- Principle 1: Break down complex problems into smaller sub-problems.
- Principle 2: Organize sub-problems to be solved independently.
- Principle 3: Solve sub-problems in a specific order.
- Principle 4: Combine sub-problem solutions to yield the original problem's solution.
Algorithmic Successes (Discrete Fourier Transform)
- Breaks down a waveform into periodic components
- Applications include DVD, JPEG, MRI, and astrophysics.
- Brute force: N² steps
- FFT algorithm: N log N steps, enabling new technologies
- Developed by Friedrich Gauss (1805)
Algorithmic Successes (N-body Simulation)
- Simulates gravitational interactions between N bodies
- Brute force: N² steps
- Barnes-Hut algorithm: N log N steps, enabling new research
- Developed by Andrew Appel (PU '81)
Why Algorithms are Useful
- Re-usable solutions to recurring problems.
- Following instructions precisely solves problems without additional thinking.
- All problem-solving knowledge is directly within the algorithm.
Why Write Algorithms Down
- Future re-use without needing to rediscover the solution
- Modification and Improvement
- Easy explanation to others
Why Analyze Algorithms
- Proving algorithms complete within a given time frame
- Identifying the fastest solution to a problem without coding and testing different algorithms
- Knowing problem complexity classes and avoiding wasted time on problems impossible on practical devices
But Computers Are So Fast These Days
- Computational demands in many problems remain
- Speed and efficiency remain necessary
Algorithms Must be Analyzed
- Classify problems and algorithms by complexity
- Predict performance and comparison of algorithms for parameter tuning
- Enhance understanding and improve implementations and algorithms
Importance of Analyzing Algorithms (1)
- Limiting various algorithms for a problem
- Understanding the relationship between problem size and running time.
- Learning about algorithm's running time without coding.
- Evaluating techniques to write efficient code and identifying bottlenecks to optimization
Importance of Analyzing Algorithms (2)
- Array-based list retrieve operation takes at most one operation, while a linked-list-based list retrieve operation takes at most N operations.
- Insert/delete are easier on linked list implementations.
- ADT implementation selection should consider frequency of operations.
- Efficiency consideration is optional for small problem sizes.
- Tradeoffs are to be considered between algorithms, time requirements and memory requirements.
What do we Analyze about Algorithms
- Algorithm behaviour and improvement
- Correctness: Matching of input/output to algorithm requirements?
- Amount of Work: Basic operations for the task
- Amount of Space: Memory usage
- Simplicity: Clear to understand.
- Verification/ Implementation
- Optimality: Is there a better solution and/or more efficient way?
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