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Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Randomized QuickSort
Consider Randomized-Quicksort operating on a sequence of \(n\) distinct input numbers. Prove that the expected running time of Randomized-Quicksort is \(O(n{\log}n)\) Prove that for any constant …
Computer Science
Algorithms
Randomized Algorithms
expectation
sorting
0
Undergraduate
By
Shiva Kintali
on Nov. 14, 2012 | Updated Dec. 6, 2017
Randomized k-SAT algorithm
Consider an instance of SAT with \(m\) clauses, where every clause has exactly \(k\) literals. Give a Las Vegas algorithm that finds an assignment satisfying at least \(m(1-2^{-k})\) clauses, and an…
Mathematics
Probability
conditional expectation
derandomization
expectation
sat
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Murphy's Law
Let \(A_1, A_2, \ldots ,A_n\) be independent events, and let \(T\) be the number of these events that occur. Show that the probability that none of the events occur is at most \(e^{-E[T]}\), wh…
Mathematics
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Runs in coin tosses
You have a biased coin i.e., each coin toss is a head with probability \(p\) and is a tail with probability \(1-p\). Let \(X\) be the number of runs in \(n\) independent tosses of this coin. Here, run…
Mathematics
Probability
expectation
variance
0
Undergraduate
By
Shiva Kintali
on June 6, 2013 | Updated Dec. 6, 2017
Two points
Two points are chosen randomly and independently from the interval \([0,1]\) according to uniform distribution. What is the expected distance between the two points ?
Mathematics
Probability
expectation
interview question
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Randomly built binary search tree
Define a randomly built binary search tree on \(n\) keys as one that arises from inserting the keys in random order into an initially empty tree, where each of the \(n!\) permutations of the input key…
Computer Science
Mathematics
Algorithms
Data Structures
Probability
binary tree
expectation
0
Undergraduate
By
Shiva Kintali
on Oct. 21, 2012 | Updated Dec. 6, 2017
Hat Check problem
Each of \(n\) customers gives a hat to a hat-check person at a restaurant. The hat-check person gives the hats back to the customers in a random order. What is the expected number of customers who g…
Mathematics
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Triangles in a random graph
Let \(G = G(n, \frac{1}{2})\) be a random graph on \(n\) vertices, i.e., for each pair of verices \(i, j\), we add the edge \((i, j)\) independently with probability \(\frac{1}{2}\). Let \(T_n\) be th…
Mathematics
Graph Theory
Probability
expectation
high probability
random graphs
triangles
variance
0
Undergraduate
By
Shiva Kintali
on Oct. 21, 2012 | Updated Dec. 6, 2017
Unbiasing a biased coin
Suppose that you want to output 0 with probability 1/2 and 1 with probability 1/2. At your disposal is a procedure BIASED-RANDOM, that outputs either 0 or 1. It outputs 1 with some probability \(p\) a…
Computer Science
Mathematics
Algorithms
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Basics of Expectation and Variance
If the random variable \(X\) takes values in non-negative integers, prove that: \(E[X] = \sum_{t=0}^\infty \Pr(X > t)\) Prove that if \(X_1\) and \(X_2\) are independent random variables then …
Mathematics
Probability
basics
expectation
variance
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