# Questions tagged [randomized-algorithms]

Questions about algorithms whose behaviour is determined not only by its input but also by a source of random numbers.

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### A so-called random variable not being well-defined

Consider this algorithm: ...
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### Some questions about RANDOM(a, b)

This is a question from CLRS: Describe an implementation of the procedure RANDOM(a, b) that only makes calls to RANDOM(0, 1). What is the expected running time of your procedure, as a function of $a$ ...
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### Yao's min-max theorem: how can the set of deterministic algorithms be finite?

I'm read Randomized Algorithms book by Motwani, the part about Yao's min-max technique: Consider a problem where the number of distinct inputs of a fixed size is finite, as is the number of distinct (...
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### Approximate duplicate sampling from a stream

The following question (in two parts) comes from a homework sheet of the fall 2019 semester cs170 course taught at UC Berkely taught by professors Vazerani and Tal. Design an algorithm that takes in ...
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### FPT algorithm for a variant of Feedback Vertex Set

I am interested in a variant of the Feedback Vertex Set (FVS) problem. For an undirected graph $G$ and $k\in \mathbb{N}$ we need to decide if there is a subset $S \subseteq V(G)$ of size at most $k$ s....
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### Deriving a lower bound on the conditional entropy, conditioned on an event

I came across Lemma 19 in Certifying Equality With Limited Interaction, which states the following for jointly distributed random variables $Z$, $W$, where $Z$ takes values in $\{0,1\}^n$, and some ...
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### Number of unique values in array in $\theta(n)$ average expected time

My idea is to initialize a hash table (with chaining) with $n$ cells, having load factor $\alpha = 1$ hence having $\theta(1)$ expected number of values in each cell in the hash table, then go cell by ...
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### Probability that two specific elements are in uniformly random sample

Consider the sampling algorithm as described here section 2.2 specifically Algorithm 2.4. Essentially we are given a stream of $N$ elements and wish to maintain a uniformly random sample, $S$, of size ...
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### What is the advantage of probability algorithm?

What is the advantage of probability algorithm? e.g. Las Vegas. I would also like to know some applications of the randomized algorithms. Are there any recommendable courses or books?
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### How to get the expected time complexity of while loop?

How to get the expected time complexity of while loop below? ...
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### How can I quickly judge whether matrix A is the inverse matrix of B?

How can I quickly judge whether matrix A is the inverse matrix of B? This is an exercise for the course I take. This question is given in the section of randomized algorithms. So I think its solution ...
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### Count Sketch probability bound

I have been reading up on the Count Sketch algorithm, and I stumpled upon the Count Sketh algorithm explained in section 5 of https://www.cs.dartmouth.edu/~ac/Teach/data-streams-lecnotes.pdf. Then, I ...
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### Quicksort: Probability of an element being compared to fewer than $k$ pivot elements

Assume we want to use quicksort on some array $s$ with length $n$ consisting of only $n$ distinct elements. Let $S_{(1)},S_{(2)},\dots,S_{(n)}$ be the sorted order of the elements in $S$. Furthermore, ...
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### Finding lowest point in circles

Given n disks in the plane, i want to compute the lowest point in their intersection area, im looking for a simple randomized incremental algorithm. There are some circles in the plane, these circles ...
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### Will random function ever hit a hard coded decimal number?

I'm not very sure how exactly a fractal number is stored nor how random function works in mordern programming languages. But I am curious, will random function ever hits a hard coded decimal number? I ...
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### k disjoint triangles with graph splitting to two distinct groups

Please note that this question is different than this question. The $k$-disjoint triangles problem is as follows: Input: A graph $G=(V,E)$ and an integer $k\in \mathbb{N}$ Output: Are there $k$ ...
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### Representation of connected components in the $O(|E|)$ time/space variant of Karger's algorithm

I'm trying to understand the various optimizations given in the original 1992 paper on Karger's algorithm. Specifically, looking at section "3.1 Unweighted Graphs", I don't understand what ...
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### Why does a polytime hitting set generator derandomize RP?

I am reading Goldreich, Vadhan, Wigderson: Simplified Derandomization of BPP Using a Hitting Set Generator and trying to understand the result that polytime hitting set generators (HSGs) would not ...
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### Randomized algorithm for minimum cut

Given a simple undirected connected graph $G$, I want to find a min-cut of $G$ using a randomized algorithm. My attempt was to select a random edge in $G$ and reduce that edge to a single vertex. And ...
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### How are randomized restarts in local search 4 times likely to give bad local minima?

I am reading section 9.3.3 Dealing with local optima in Algorithms by Dasgupta et al. and the authors mention that in randomized restarts, it is four times likely to end up with a bad solution. They, ...
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### What are the advantages of using PRNG over TRNG?

True random number generators use an unpredictable physical means to generate numbers, whereas pseudo-random numbers utilize mathematical formulas to produce a certain sequence of numbers that will ...
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### can a machine generate truly random numbers?

I know that for most programs pseudo-random numbers are sufficient, but there are ways that machines can generate truly random numbers! There are devices that generate unpredictable processes. However,...
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### Scott Aaronson's Proof of $\textbf{BPP} \subset \textbf{P/poly}$

The proof is in the image below, taken from "Quantum Computing Since Democritus": Here's what I don't totally get: my understanding of random algorithms is that randomization is not done ...
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### Question on an Algorithm for Longest Increasing Subsequence

I have been reading this paper: https://arxiv.org/abs/2011.10874 This paper presented an exact randomized algorithm with update time $\tilde{O}(n^{0.8})$. I will quickly talk about the overall idea of ...
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### Proving an algorithm satisfies $\epsilon$-DP

I am trying to prove/disprove that an algorithm satisfies $\epsilon$-DP. I proved some of them but there are 3 more which I could not decide on. Here they are: Algorithm A takes as input a dataset D ...
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### Algorithmic challenge: generate a list of random non overlapping squares

For an undisclosed reason, I need a list of $n$ squares in a two dimensions space where each square does not overlap. So the challenge is simply: given a two dimensional area $a$ (...
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### 2-Approximation algorithm for for messages across a cyclic network

Question There are $n$ computers arranged in a cycle ($1,2,3..,n,1$), with undirected edges between adjacent computers. There are $m$ messages that need to be delivered. Message $i$ ($1 \le i \le m$) ...
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### Question in coreset construction fro K-median clustering

I was reading Ke chen's paper about coreset construction for K-median clustering. In this paper, he assumed that $A$ is an $[α, β]$-bicriteria approximation for K-median clustering for some $α, β=O(1)$...
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### Definition of BPP

We know that BPP is described as $\{L\mid \exists \text{ TM }M, \text{ s.t. }\Pr[M(x)=L(x)]\geq2/3\}$. I saw a proof which uses Chernoff bound to prove that any probability larger than $1/2$ can be ...
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### Algorithm for cyclic $n$-string Hamming distance with constant sized language $\Sigma$

Suppose we are given a language $\Sigma$ where, suppose, $|\Sigma| = O(1)$. Consider two fixed strings $A, B \in \Sigma^n$. Define the Hamming metric between these strings as d_{H}(A,B) = \sum_{i=1}^...
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### Let M be a k × n random matrix with iid entries such that

$M$ is a $k × n$ random matrix with iid entries such that $P(m_{i,j} = +1) = P(m_{i,j} = −1) = 0.5.$ Let $k = O({1\over \epsilon^l})$ for some constant $l$. $v ∈ R_n$ is a fixed vector. Does a ...
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### The observation of the coreset in K-median clustering problems [duplicate]

I have seen two observations from the paper by Har-Peled but I do not know how to prove them (i) If $C1$ and $C2$ are the $(k, ε)$-coresets for disjoint sets P1 and P2 respectively, then $C1 ∪ C2$ is ...
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### About the properties of the coresets in k-median clustering

I have seen two observations from the paper by Har-Peled but I do not know how to prove them (i) If $C1$ and $C2$ are the $(k, ε)$-coresets for disjoint sets P1 and P2 respectively, then $C1 ∪ C2$ is ...
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### Existence of Pseudorandom Generator

How to show that for $\epsilon>0$, there exists a function $G:\{0,1\}^n->\{0,1\}^{2^{\epsilon n}}$ that is a $2^{\epsilon n}$-prg, without the condition that is is computable in $2^{O(n)}$ time. ...
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### derandomize a BPP algorithm

Suppose we have a BPP algorithm $A$ s.t. its running time is random and is $O(n^2)$ in expectation. How do we create a new BPP algorithm $B$ to solve the same problem s.t. it has deterministic running ...
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### Generate a uniform random numbers in $O(1)$

Suppose you have access to a random number generator $G()$ that generates uniform random numbers in $\{0,\cdots,n-1\}$. (Here, $n$ is given and cannot be changed.) How do we generate a uniform random ...
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### Randomized Assignment Problem

Given $x_1,...,x_n,y_1,...,y_n\in \mathbb{R}^d$ find a permutation matrix $P\in\mathbb{S}_d$ that minimizes $\sum_{ij}P_{ij}|x_i-y_j|$. This is an assignment problem and can be solved in $O(n^3+n^2d)$ ...
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### Maximization problem

I work at a company and i got to a seminar we're they told us to solve this problem below in the picture Is there an algorithm that can help me solve this question. I thought about a randomized ...
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### Randomly generating graph based off number of connections on each node

I'm trying to generate a graph based off some data I have. This graph should have N nodes where the number of edges each node has is equal to a random number ...
### Efficiently selecting a random subset of size $m$ from a set of size $n$
This is a cross post of my question here on math.se. I have a list of $n$ items and would like to randomly select an $m$ set from it efficiently (in terms of time complexity). Also, I want all ...
Consider some problem $P$ and let's assume we sample the problem instance u.a.r. from some set $I$. Let $p$ be a lower bound on the distributional error of a deterministic algorithm on $I$, i.e., ...