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Questions and Answers
What is an algorithm?
What is an algorithm?
An algorithm is a step-by-step procedure or formula for solving a problem.
What do you mean by correct algorithm?
What do you mean by correct algorithm?
A correct algorithm produces the correct output for all valid inputs.
What do you mean by instance of a problem?
What do you mean by instance of a problem?
An instance of a problem is a specific input to which the algorithm will be applied.
List out the criteria that all algorithms must satisfy. (Select all that apply)
List out the criteria that all algorithms must satisfy. (Select all that apply)
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On what basis will you consider algorithm A is better than algorithm B?
On what basis will you consider algorithm A is better than algorithm B?
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What is an amortized analysis?
What is an amortized analysis?
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Explain accounting method and aggregate analysis with suitable example.
Explain accounting method and aggregate analysis with suitable example.
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What is a set?
What is a set?
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What is a relation?
What is a relation?
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What is a function?
What is a function?
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Calculate computation time for the statement 't3' in the following code fragment: for i = 1 to n { for j = 1 to i { c = c + 1 ..... t3 }}
Calculate computation time for the statement 't3' in the following code fragment: for i = 1 to n { for j = 1 to i { c = c + 1 ..... t3 }}
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Prove that T(n) = 1 + 2 + 3 + ... + n = Θ(n^2).
Prove that T(n) = 1 + 2 + 3 + ... + n = Θ(n^2).
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Define an amortized analysis.
Define an amortized analysis.
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Explain the key components of Big 'Oh' notation.
Explain the key components of Big 'Oh' notation.
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What is Recursion?
What is Recursion?
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Explain the Tower of Hanoi Problem.
Explain the Tower of Hanoi Problem.
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What are the general characteristics of Greedy Algorithms?
What are the general characteristics of Greedy Algorithms?
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What is the difference between Greedy Algorithms and Dynamic Programming?
What is the difference between Greedy Algorithms and Dynamic Programming?
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Study Notes
Unit 1: Basics of Algorithms and Mathematics
- Definition of Algorithm: A precise sequence of instructions to solve a specific problem or perform a computation.
- Correct Algorithm: An algorithm that effectively solves the intended problem and produces the correct output for all valid inputs.
- Instance of a Problem: A specific input scenario for which an algorithm is executed, representing a single case within the problem domain.
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Criteria for Algorithms:
- Finiteness: Must terminate after a finite number of steps.
- Definiteness: Each step must be precisely defined.
- Effectiveness: Each operation can be carried out by a person using a pencil and paper.
- Input: Should have zero or more inputs.
- Output: Should produce one or more outputs.
- Comparing Algorithms: An algorithm A is considered better than algorithm B based on factors like time complexity, space complexity, ease of implementation, and the size of input it can handle efficiently.
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Linear Inequalities and Equations:
- Linear equations represent the relationship between variables using a straight line (e.g., ax + b = 0).
- Linear inequalities express a range of values (e.g., ax + b < c).
Asymptotic Notation
- Asymptotic Notation: Used to describe the behavior of functions as inputs approach infinity, indicating how the runtime or space requirements grow.
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Common Asymptotic Notations:
- Big O (O): Upper bound on the time (or space) complexity.
- Big Omega (Ω): Lower bound on the time (or space) complexity.
- Theta (Θ): Tight bound on the time (or space) complexity, meaning it serves as both upper and lower bound.
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Master Theorem: Provides a method to analyze the time complexity of divide-and-conquer algorithms.
- Example Recurrence Equations:
- T(n) = 9T(n/3) + n
- T(n) = 3T(n/4) + n log n
- T(n) = T(2n/3) + 1
- Example Recurrence Equations:
- Amortized Analysis: Analyzes the average time taken per operation over a sequence of operations, smoothing out worst-case scenarios.
- Techniques of Amortized Analysis: Include accounting method and aggregate analysis.
Algorithm Analysis and Complexity
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Complexity Categories:
- Worst Case: Maximum time required by an algorithm for the worst input size.
- Best Case: Minimum time required for the best input.
- Average Case: Expected time for a random input.
- Recursion: A method where a function calls itself, and is often used in algorithms like the Tower of Hanoi.
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Sorting Algorithms:
- Bubble Sort: Simple comparison-based sorting technique, analyzed for best, worst, and average case complexities.
- Selection Sort: Selects the minimum element and swaps it, analyzed for best, worst, and average complexities.
- Heap Sort: A comparison-based sorting algorithm using a binary heap. Complexity to be reviewed in detail.
Key Concepts and Terms
- Quantifier: Used in logic to specify the quantity of specimens in the domain of discourse (e.g., "for all" or "there exists").
- Recursion Equation: Expresses the relationship in recursive algorithms; for instance, Tower of Hanoi's recursion can illustrate its complexity.
- Analysis Importance: Understanding algorithm performance helps in choosing the most efficient algorithms for specific tasks.
Unit 2: Divide and Conquer Algorithms
- Greedy Algorithms: Often yield good solutions quickly but do not guarantee optimality; contrasted with dynamic programming and divide-and-conquer which design more comprehensive strategies.
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Algorithm Characteristics:
- Greedy focuses on local optimums, while dynamic programming considers global optimum.
- Divide and conquer breaks problems into subproblems, solves them independently, then combines results.
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Description
This quiz covers the fundamentals of algorithms, including definitions and the concept of a correct algorithm. Explore key terms and principles that are essential for understanding the analysis and design of algorithms. Perfect for students in the 5th semester of their academic journey.