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# 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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2answers
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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 ...
1answer
44 views

### 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?
2answers
33 views

### How to get the expected time complexity of while loop?

How to get the expected time complexity of while loop below? ...
3answers
2k views

### 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 ...
1answer
146 views

### 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 ...
2answers
36 views

### 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, ...
1answer
34 views

### 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 ...
2answers
29 views

### 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 ...
2answers
147 views

1answer
77 views

### 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 $k$ ...
0answers
18 views

### 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 ...
1answer
187 views

### 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 ...
1answer
35 views

### 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 ...
0answers
12 views

### 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, ...
1answer
121 views

### 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 ...
1answer
79 views

### 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,...
1answer
38 views

### 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 ...
2answers
64 views

1answer
64 views

### 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 ...
0answers
21 views

### 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 ...
1answer
132 views

### 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 $a$ (...
1answer
54 views

### 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 $i$ ($1 \le i \le m$) ...
1answer
33 views

### 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 $α, β=O(1)$...
1answer
27 views

### 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 ...
1answer
44 views

### 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}^...
0answers
31 views

### 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 ...
0answers
21 views

### 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 ...
2answers
71 views

### 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 ...
1answer
44 views

### 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. ...
1answer
76 views

### 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 ...
1answer
37 views

### 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 ...
0answers
42 views

### 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 $O(n^3+n^2d)$ ...
2answers
49 views

### 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 ...
1answer
22 views

### 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 ...
1answer
186 views

### 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 ...
0answers
27 views

### Distributional error probability of deterministic algorithm implies error probability of randomized algorithm?

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., ...
1answer
31 views

### Median of distribution with memory constraint

Task I want to approximate the median of a given distribution $D$ that I can sample from. A simple algorithm for this, using $n$ samples, is: ...
3answers
255 views

### Book recommendations on the analysis of randomized algorithms

I would like to read some books (or any other material) that cover the design of randomized algorithms with a particular focus on the analysis. My main goal is to develop the rigour needed to ...
1answer
194 views

### Weighted Online Matching - randomized algorithms

Let's consider the edge weighted online matching problem. The Vertices arrive online and reveal all their current edges and edge-weights $w_e>0$. The goal is to maximize the matchings weight. An ...
0answers
22 views

### Analyzing a counting triangles streaming algorithm which uses $\ell_0$ sampling

I'm trying to analyze the following streaming algorithm for counting triangles (see below). It supposedly works also for dynamic graphs (i.e. "turnstile model", where edge deletions are ...
1answer
124 views

### Is any randomized Algorithm a probability distribution over the set of deterministic Algorithms?

If there is a finite set of Instances of size n and the set of (reasonable) deterministic algorithms is finit. Can any randomized Algorithm be seen as a probability distribution over the set of ...
1answer
154 views

### Randomized Algorithms: High-Probability vs. Expectation

Hopefully this question isn't too general, but I was wondering what the relationship is between randomized algorithms that perform well with high-probability and those that perform well in expectation....
0answers
25 views

### Space complexity of using a pairwise independent hash family

I'm trying to analyze the space complexity of using the coloring function $f$ which appears in "Colorful Triangle Counting and a MapReduce Implementation", Pagh and Tsourakakis, 2011, https:...