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)

<p>Finiteness</p> Signup and view all the answers

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

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

What is an amortized analysis?

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

Explain accounting method and aggregate analysis with suitable example.

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

What is a set?

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

What is a relation?

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

What is a function?

<p>A function is a relation that uniquely associates each element of a set with exactly one element from another set.</p> 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 }}

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

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

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

Define an amortized analysis.

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

Explain the key components of Big 'Oh' notation.

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

What is Recursion?

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

Explain the Tower of Hanoi Problem.

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

What are the general characteristics of Greedy Algorithms?

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

What is the difference between Greedy Algorithms and Dynamic Programming?

<p>Greedy algorithms make decisions based on local optimal solutions, while dynamic programming considers global optimal solutions.</p> 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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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.

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