Try it on GfG Practice Given a number N, generate bit patterns from 0 to 2^N-1 such that successive patterns differ by one bit. Examples: Input: N = 2 Output: 00 01 11 10 Input: N = 3 Output: 000 001 011 010 110 111 101 100 Method-1 The above sequences are Gray Codes of different widths. Following…

Given three integers a, b, and c, the task is to find the smallest integer without using any comparison operators. Examples: Input: a = 2, b = 5, c = 7 Output: 2 Explanation: The smallest of the 3 integers is a, which is equal to 2. Input: a = 14, b = 13, c = 10 Output: 10 Explanation: The smallest…

We strongly recommend to refer below post as a prerequisite of this. Randomized Algorithms | Set 1 (Introduction and Analysis) Classification Randomized algorithms are classified in two categories. Las Vegas: A Las Vegas algorithm were introduced by Laszlo Babai in 1979. A Las Vegas algorithm is an…

What is a Randomized Algorithm? An algorithm that uses random numbers to decide what to do next anywhere in its logic is called a Randomized Algorithm. For example, in Randomized Quick Sort, we use a random number to pick the next pivot (or we randomly shuffle the array). And in Karger's algorithm,…

How many people must be there in a room to make the probability 100% that at-least two people in the room have same birthday? Answer: 367 (since there are 366 possible birthdays, including February 29). The above question was simple. Try the below question yourself. How many people must be there in…

Given a function rand2() that returns 0 or 1 with equal probability, implement rand3() using rand2() that returns 0, 1 or 2 with equal probability. Minimize the number of calls to rand2() method. Also, use of any other library function and floating point arithmetic are not allowed. The idea is to…

Given a function rand50() that returns 0 or 1 with equal probability, write a function that returns 1 with 75% probability and 0 with 25% probability using rand50() only. Minimize the number of calls to the rand50() method. Also, the use of any other library function and floating-point arithmetic…

You are given a function rand(a, b) which generates equiprobable random numbers between [a, b] inclusive. Generate 3 numbers x, y, z with probability P(x), P(y), P(z) such that P(x) + P(y) + P(z) = 1 using the given rand(a,b) function.The idea is to utilize the equiprobable feature of the rand(a,b)…

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Given an array of integers, find the most occurring element of the array and return any one of its indexes randomly with equal probability.Examples: Input: arr[] = [-1, 4, 9, 7, 7, 2, 7, 3, 0, 9, 6, 5, 7, 8, 9] Output: Element with maximum frequency present at index 6 OR Element with maximum…

A CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart) is a test to determine whether the user is human or not.So, the task is to generate unique CAPTCHA every time and to tell whether the user is human or not by asking user to enter the same CAPTCHA as generated…

Given a deck of cards, the task is to shuffle them. Asked in Amazon Interview Prerequisite : Shuffle a given array Algorithm: 1. First, fill the array with the values in order. 2. Go through the array and exchange each element with the randomly chosen element in the range from itself to the end. //…

Expectation or expected value of any group of numbers in probability is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up in an extremely large number of…

Consider the following famous puzzle. In a country, all families want a boy. They keep having babies till a boy is born. What is the expected ratio of boys and girls in the country? This puzzle can be easily solved if we know following interesting result in probability and expectation. If…

Given an array, write a program to generate a random permutation of array elements. This question is also asked as "shuffle a deck of cards" or "randomize a given array". Here shuffle means that every permutation of array element should be equally likely. Let the given array be arr[]. A simple…

You are given a function foo() that represents a biased coin. When foo() is called, it returns 0 with 60% probability, and 1 with 40% probability. Write a new function that returns 0 and 1 with a 50% probability each. Your function should use only foo(), no other library method. Solution: We know…

We are given a sorted array A[] of n elements. We need to find if x is present in A or not.In binary search we always used middle element, here we will randomly pick one element in given range.In Binary Search we had middle = (start + end)/2 In Randomized binary search we do following Generate a…

Given a function, rand6() that returns random numbers from 1 to 6 with equal probability, implement the one-liner function rand12() using rand6() which returns random numbers from 1 to 12 with equal probability. The solution should minimize the number of calls to the rand6() method. Use of any other…

Monte Carlo estimation Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. One of the basic examples of getting started with the Monte Carlo algorithm is the estimation of Pi. Estimation of Pi The idea is to simulate…

Given two sparse matrices (Sparse Matrix and its representations | Set 1 (Using Arrays and Linked Lists)), perform operations such as add, multiply or transpose of the matrices in their sparse form itself. The result should consist of three sparse matrices, one obtained by adding the two input…

In this article, we will discuss how to implement QuickSort using random pivoting. In QuickSort we first partition the array in place such that all elements to the left of the pivot element are smaller, while all elements to the right of the pivot are greater than the pivot. Then we recursively call…

Given a password entered by the user, check its strength and suggest some password if it is not strong. Criteria for strong password is as follows : A password is strong if it has : At least 8 characters At least one special char At least one number At least one upper and one lower case char.…

Reservoir sampling is a family of randomized algorithms for randomly choosing k samples from a list of n items, where n is either a very large or unknown number. Typically n is large enough that the list doesn't fit into main memory. For example, a list of search queries in Google and Facebook. So…

Given n numbers, each with some frequency of occurrence. Return a random number with probability proportional to its frequency of occurrence. Example: Let following be the given numbers. arr[] = {10, 30, 20, 40} Let following be the frequencies of given numbers. freq[] = {1, 6, 2, 1} The output…

Given a stream of numbers, generate a random number from the stream. You are allowed to use only O(1) space and the input is in the form of a stream, so can't store the previously seen numbers. So how do we generate a random number from the whole stream such that the probability of picking any…

Given a function random01Generator() that gives you randomly either 0 or 1, implement a function that utilizes this function and generate numbers between 0 and 6(both inclusive). All numbers should have same probabilities of occurrence. Examples: on multiple runs, it gives 3 2 3 6 0 Approach : The…

Given a function foo() that returns integers from 1 to 5 with equal probability, write a function that returns integers from 1 to 7 with equal probability using foo() only. Minimize the number of calls to foo() method. Also, use of any other library function is not allowed and no floating point…

Given a singly linked list, select a random node from the linked list (the probability of picking a node should be 1/N if there are N nodes in the list). You are given a random number generator.Below is a Simple Solution Count the number of nodes by traversing the list. Traverse the list again and…

Given a Binary Tree with children Nodes, Return a random Node with an equal Probability of selecting any Node in tree.Consider the given tree with root as 1. 10 / \ 20 30 / \ / \ 40 50 60 70 Examples: Input : getRandom(root); Output : A Random Node From Tree : 3 Input : getRandom(root); Output : A…

Prerequisite: Random Variable This post is about mathematical concepts like expectation, linearity of expectation. It covers one of the required topics to understand Randomized Algorithms. Let us consider the following simple problem. Problem: Given a fair dice with 6 faces, the dice is thrown n…

Given two integers N and M, the task is to generate any N * M sized maze containing only 0 (representing a wall) and 1 (representing an empty space where one can move) with the entry point as P0 and exit point P1 and there is only one path between any two movable positions. Note: P0 and P1 will be…

Given three matrices A, B and C, find if C is a product of A and B. Examples: Input : A = 1 1 1 1 B = 1 1 1 1 C = 2 2 2 2 Output : Yes C = A x B Input : A = 1 1 1 1 1 1 1 1 1 B = 1 1 1 1 1 1 1 1 1 C = 3 3 3 3 1 2 3 3 3 Output : No A simple solution is to find product of A and B and then check if…

Haskell is a high-level, safe programming language often misunderstood as lacking performance when compared to lower-level languages like C and Rust. Many believe Haskell is unsuitable for systems programming due to these performance concerns. In this talk, we will present a real-world use case that…

Quicksort is a well known algorithm for sorting, Professor Graham Hutton shows how it works and then how to implement it in just five lines of code! EXTRA BITS: https://youtu.be/7cEaRVEz__I Professor Hutton is based at the University of Nottingham: https://people.cs.nott.ac.uk/pszgmh/ Computerphile…

[James Bruton] is experimenting is a series of interesting mechanical mechanisms, the latest being a CVT transmission system which uses a tilting sphere to get a variable speed output from a constant speed input. Video after the break. In [James]’ proof of concept RC vehicle, a single powered disc…

Check out the new video from Oxford Social Sciences, where Max shares his journey with this project. January 30, 2025 The University of Oxford’s Social Sciences Division has featured our founder, Max Roser, in a short video about Our World in Data as part of their Innovation Stories series. In the…

This feature first appeared in April 2024 on Londonist: Time Machine, our much-praised history newsletter. To be the first to read new history features like this, sign up for free here. Walk around Soho and Covent Garden and one name pops up time and time again. John Logie Baird, Scottish-born…