Randomized Algorithms

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What is Randomized Algorithm?

An algorithm that uses random numbers to decide what to do next anywhere in its logic is called Randomized Algorithm.

For example, in Randomized Quick Sort, we use a random number to pick the next pivot (or we randomly shuffle the array). Typically, this randomness is used to reduce time complexity or space complexity in other standard algorithms.

Topics :

Introduction:

1. Random Variables
2. Binomial Random Variables
3. Randomized Algorithms | Set 0 (Mathematical Background)
4. Randomized Algorithms | Set 1 (Introduction and Analysis)
5. Randomized Algorithms | Set 2 (Classification and Applications)
6. Randomized Algorithms | Set 3 (1/2 Approximate Median)

Problems on Randomized Algorithms:

• Easy:
  1. Write a function that generates one of 3 numbers according to given probabilities
  2. Generate 0 and 1 with 25% and 75% probability
  3. Implement rand3() using rand2()
  4. Birthday Paradox
  5. Expectation or expected value of an array
  6. Shuffle a deck of cards
  7. Program to generate CAPTCHA and verify user
  8. Find an index of maximum occurring element with equal probability
  9. Randomized Binary Search Algorithm
• Medium:
  1. Make a fair coin from a biased coin
  2. Shuffle a given array using Fisher–Yates shuffle Algorithm
  3. Expected Number of Trials until Success
  4. Strong Password Suggester Program
  5. QuickSort using Random Pivoting
  6. Operations on Sparse Matrices
  7. Estimating the value of Pi using Monte Carlo
  8. Implement rand12() using rand6() in one line
• Hard:
  1. Generate integer from 1 to 7 with equal probability
  2. Implement random-0-6-Generator using the given random-0-1-Generator
  3. Select a random number from stream, with O(1) space
  4. Random number generator in arbitrary probability distribution fashion
  5. Reservoir Sampling
  6. Linearity of Expectation
  7. Introduction and implementation of Karger’s algorithm for Minimum Cut
  8. Select a Random Node from a Singly Linked List
  9. Select a Random Node from a tree with equal probability
  10. Freivald’s Algorithm to check if a matrix is product of two
  11. Random Acyclic Maze Generator with given Entry and Exit point

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