Algorithms Basics - Question Bank Semester 5

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Questions and Answers

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?

A correct algorithm produces the correct output for all valid inputs.

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)

Finiteness Signup and view all the answers

On what basis will you consider algorithm A is better than algorithm B?

Algorithm A is considered better than algorithm B based on time complexity, space complexity, and efficiency. Signup and view all the answers

What is an amortized analysis?

Amortized analysis is a method of analyzing the running time of an algorithm over a sequence of operations. Signup and view all the answers

Explain accounting method and aggregate analysis with suitable example.

The accounting method assigns a cost to operations, while aggregate analysis calculates the total cost for a series of operations. Signup and view all the answers

What is a set?

A set is a collection of distinct objects considered as a whole. Signup and view all the answers

What is a relation?

A relation is a set of ordered pairs of elements. Signup and view all the answers

What is a function?

A function is a relation that uniquely associates each element of a set with exactly one element from another set. Signup and view all the answers

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 }}

The computation time for statement 't3' is O(n^2). Signup and view all the answers

Prove that T(n) = 1 + 2 + 3 + ... + n = Θ(n^2).

The sum of the first n integers is T(n) = n(n+1)/2, which simplifies to Θ(n^2). Signup and view all the answers

Define an amortized analysis.

Amortized analysis evaluates the time complexity of operations over a sequence, providing average costs. Signup and view all the answers

Explain the key components of Big 'Oh' notation.

Big 'Oh' notation describes the upper bound of an algorithm's time complexity. Signup and view all the answers

What is Recursion?

Recursion is a method where a function calls itself to solve smaller instances of the same problem. Signup and view all the answers

Explain the Tower of Hanoi Problem.

The Tower of Hanoi is a mathematical puzzle involving three rods and a number of disks of different sizes. Signup and view all the answers

What are the general characteristics of Greedy Algorithms?

Greedy algorithms build up a solution piece by piece, always choosing the next piece that offers the most immediate benefit. Signup and view all the answers

What is the difference between Greedy Algorithms and Dynamic Programming?

Greedy algorithms make decisions based on local optimal solutions, while dynamic programming considers global optimal solutions. Signup and view all the answers

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.
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.
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.
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.
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
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

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.
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.
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.