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Questions tagged [probabilistic-algorithms]

Questions about (typically randomized) algorithms that can produce no or an incorrect answer with a certain probability.

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8 views

From coin flips to algebraic functions via pushdown automata

Given a coin with probability of heads of $\lambda$, sample the probability $f(\lambda)$. This is the Bernoulli factory problem, and it can be solved only for certain functions $f$. (For example, ...
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1answer
34 views

Obtaining an expectation in uniform hashing

long shot question but I am super stuck. Donald Knuth has proven (p. 8 here, equation 12) that the probability that the maximum value in uniform hashing is smaller than $n/2$ is equal to 0.288. I ...
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1answer
20 views

Property testing of a complete multipartite graph

Propose and prove an $\epsilon$-test for the following property in the dense graph model: $G=(V,E)$ is a complete multipartite graph. That is, there exists a partition $V=V_1\cup\ldots\cup V_\ell$ ...
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1answer
24 views

Dynamic programming and probability - list of problems

Does anyone have a list of problems where you have to combine dynamic programming with probability?
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1answer
16 views

How to sample Bivariate Normal Distribution with Accept reject method

I have to write python code in jupyter due to sampling bivariate normal distribution with 3 sampling methods: Prior Sampling Gibbs Sampling Rejection Sampling I have done the first two samplings and ...
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0answers
61 views

Noisy sorter: optimal algorithm

Given $n$ elements $x_1,\dots, x_n$, and algorithm $A$ which outputs $(r_1,\dots, r_n)=A(x_1,\dots, x_n)$, we say $A$ is $\epsilon$-sorting the list if $rank(x_i) \in (r_i-n\epsilon, r_i+ n\epsilon) $ ...
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1answer
24 views

Match unpaired points between two sets

I have two sets, let's call them C and T, of unpaired points, which could for example represent two types of cells. Hence, both points are drawn from the same underlying distribution, but the points ...
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0answers
16 views

Predictive score ? how?

let say I have a sequence of values ... The values X and Y can be either number or data structures or states ... and so on ... Example: Value X can be followed by any of "n" other values Y1,...
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2answers
55 views

Probabilistic adaptive algorithm that can explore only $k$ bits of input can’t distinguish $k$-independent distribution from uniform

Definition: Distribution $D$ on $\{0,1\}^n$ is called $k$-independent if for every random variable $X$ with distribution $D$ and for all $i_1, \dots, i_k \in \{1,2,\dots,n\}$ random variable $X_{i_1,\...
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2answers
64 views

Why is tabulated hashing 3-wise independent but not 4-wise independent?

Tabulated hashing uses tables of random numbers to compute hash values. Suppose $|\mathcal{U}| = 2^w \times 2^w$ and $m = 2^l$, so that the items being hashed are pairs $(x,y)$ where $x$ and $y$ are $...
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1answer
42 views

Is there a concept of probabilistic quantum computers?

Answering my question Yonatan N said a statement from which follows that there are computable functions of quantum time complexity strictly above polynomial. Accordingly a Quora answer Quantum ...
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1answer
32 views

How to simulate online matching algorithms (implementation)

I was reading about online algorithms and bipartite matching. I found an implementation that works fine on several websites (like geeksforgeeks). For the online version, I found this paper https://...
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31 views

Techniques to prove lower bounds on randomized algorithms

I am interested in proving lower bounds for AM-like languages. The usual techniques for lower bounds in non-probabilistic machines don't work for probabilistic machines. Intuitively, when I think ...
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0answers
46 views

kth smallest element using Randomized select

I have recently started studying Randomized algorithms on my own. I am refering to Rajiv motwani - randomized algorithms book. Objective - find kth smallest element using radomized select in $O(n^\...
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1answer
30 views

Find dominated or subsumed linear inequalities efficiently

Problem statement Given a set of $N$ linear inequalities of the form $a_1x_1 + a_2x_2 + ... + a_Mx_M \geq RHS$, where $a_i$ and $RHS$ are integers. The inequality $A$ dominates or subsumes inequality $...
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1answer
38 views

How to implement conditional probability distribution on set-valued Random Variables

I'm trying to implement conditional probability distribution when the events of two RVs are sets. If I try to extrapolate concepts from real or categorical variables to sets things become confusing ...
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1answer
50 views

Calculating a jackpot winner based on probabilities

Imagine a jackpot where users can bet as much as they want, and each bet increases their winning chance. Given a roll [0-100], how would you calculate the winner? ...
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1answer
40 views

Probability of loss using a binary symmetric channel

Today we talked about Information Theory and the binary symetric channel. For newbies here is a little explanation : For instance if I want to send a binary to someone : The bit will be "flipped&...
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1answer
14 views

Populating a vector of numbers to expose an error in a function implementation

So lets say I'm writing an algorithm that takes a vector as input. I want to know that I'm writing this algorithm correctly however so I of course write tests to see if the output equals what I expect ...
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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., ...
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1answer
29 views

Hash function, $h(k) = \lfloor km \rfloor$ is simple uniform for real $k$ independently, uniformly distributed in the range $0 \leq k < 1$

I was going through the text Introduction to Algorithms by Cormen et. al. where I came across the following statement: If the keys are known to be random real numbers $k$ independently and uniformly ...
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1answer
44 views

Confusion about the Hiring Problem

I'm confused about where the probability from the hiring problem comes from. For background: We interview one person everyday who has a quality characteristic, x, from 0 to 1(distributed uniformly). ...
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1answer
33 views

Why Convolutional codes is easy to factor/handle the uncertainty of a bit being a 0 or a 1 into the decoding?

Here is an excerpt from Andrew S. Tanenbaum, Computer Networks, 5th edition, Chapter 3 (The data link layer), Page 208: My question is about this part. [Convolutional codes have been popular in ...
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1answer
154 views

What is an example of a Monte-Carlo algorithm for finding a Hamiltonian path?

I've recently been made aware that there exist Monte-Carlo algorithm(s?) for determining whether a Hamiltonian path exists in a graph. I am struggling to figure out how it would work. What is the ...
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45 views

Hashing algorithm which minimizes distribution

Consider applications A1, A2 .. AN having properties P11, P12,... PN1, PN2.. Also consider buckets ...
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1answer
55 views

Sizing a Cuckoo filter for a subset of elements

Question I am considering the use of Cuckoo filter for a business case. To simplify the explanation here is an analogy of my needs: There are over $n = 30 000$ first names that exists in the whole ...
2
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1answer
108 views

$k$-coloring in BPP, implies $k$-coloring in ZPP

Consider the next problem: $k$-COL: Given a graph $G=(V,E)$, does it have a valid $k$-coloring? I need to prove that if $k$-COL is in BPP, then it is also in ZPP. In other words, show that if there ...
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1answer
59 views

Is there a complexity class QPP?

The complexity class PP is not considered tractable, because the probability of success can get arbitrarily close to 50% from above as the problem instances get larger, so that (e.g. if this approach ...
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1answer
27 views

Recover a matrix with minimum number of queries

Alice has a matrix $A \in \{0,1\}^{n \times m}$ such that the sum of each row is $1$. Bob tries to find the indices of the ones (he knows that the sum of each row is $1$). The type of questions Bob ...
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1answer
61 views

Probability of terminating in a state in a probabilistic algorithm

Suppose i have a circular array of $n$ elements. At time $t=0$ i am in position 0 of the array. The algorithm moves left or right with probability $p=1/2$ (since the array is circular when it moves ...
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1answer
31 views

Demonstrating that probability for every possible result is uniform at the end of an algorithm

I have memory of $k$ elements that you can imagine being represented by an array. One by one, the array receives a value corresponding to the time index, for example at $t=1$ the value will be $1$. At ...
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25 views

Chernoff Bounds (upper tail)

For the proof of Chernoff Bounds (upper tail) we suppose δ<2e−1 . Like in this paper ([see this link ]) 1. Can you tell me why ?
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73 views

Prove an estimator

Consider an undirected graph $G=(V,E)$ representing the social network of friendship/trust between students. We would like to form teams of three students that know each other. The question is to ...
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21 views

Find expectation and calculate Chernoff bound [duplicate]

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same 2 tasks to every ...
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1answer
316 views

Find expectation with Chernoff bound

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same $2$ tasks to every ...
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1answer
615 views

Derandomization of vertex cover algorithm

I have the following randomized-algorithm for the vertex cover problem. Let $B_0$ be the output set: Fix some order $e_1, e_2,...,e_m$ over all edges in the edge set E of G, and set $B_0=∅$. Add to ...
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1answer
108 views

Grover's algorithm on probabilistic classical machines

As a starting point for this question, I came across this question, which AIUI is citing a construction showing how to simulate quantum circuits with a $PP$ algorithm, i.e. implying quantum ...
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1answer
16 views

Predicting the outcome of sporting events with multiplicative scoring

In the Olympic format for sport climbing, eight athletes compete in three rounds of climbing. Their final score is the multiplication of their rankings in each round. For example, an athlete who comes ...
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1answer
68 views

Understanding simulated annealing information theoretically

So I recently rediscovered simulated annealing though a path that others seem to be well aware of. I was aware of Metropolis-Hastings as a sampling algorithm that creates a Markov-Chain who's ...
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17 views

What is the probability that an expanding bipartite graph exists with the property, |V1|=|V2|?

I want to find a bound on the above problem, and show that a random graph has a positive probability of being an expanding bipartite with the property, |V1|=|V2|. I am not getting, where should I ...
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1answer
199 views

Why an error probability of 1/3 in BPP?

BPP is defined as the class of polynomial-time random algorithms which have an error probability of at most 1/3. But why was 1/3 chosen? If we have an algorithm with some error probability less than ...
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1answer
50 views

Are there efficient probabilistic multiplication algorithms that use O(n log n) gates?

Recently Harvey and Hoeven published a paper proving that integer multiplication can be performed using at most O(n log n) operations. This algorithm is theoretically interesting, but in practice ...
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1answer
38 views

Polysize bounded depth circuit for modified MAJORITY problem

I am trying to show the existence of a polynomial size, bounded depth monotone circuit on the inputs $(x_1,\ldots, x_n)$ that gives $1$ if $\sum x_i \geq n/2 + n/\log n$ and $0$ if $\sum x_i \leq n/2 -...
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1answer
157 views

Can BPP be bounded around any constant other than 1/2?

A language $L$ is in BPP if there exists a randomised TM such that it outputs a correct answer with probability at least $1/2+1/p(n)$ for some polynomial $p(n)$, where $n$ is the length of the input. ...
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0answers
77 views

Correctness of Karger's min-cut Algorithm

tl;dr in the analysis for Karger's min-cut, the probability of an edge being in the min-cut in the $j$th iteration, $\frac{k}{0.5k(n-j)}$, neglects the fact that all the edges between the two ...
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0answers
34 views

Construction of hash function with a given distribution

Two questions about the construction of a hash function: Let $U = \{u_1,...,u_n\}$ be a set of size $n$, and suppose that one is interested in a function $h\colon U \rightarrow [0,1]$ such that $h$ "...
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0answers
32 views

PCP variant in P with non 0 randomness and polynomial proof

I am trying to show that a particular language $L$ in PCP(log,q) is also in P. The PCP protocol works as follows: log many random bits and checks at q positions in a polynomial length proof. The ...
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1answer
37 views

SVM with a priori information about class probabilities

Given are two 2-d sets, each with its own bivariate normal distribution. I need to build an SVM classifier. The a priori probabilities of each class corresponds to the size of its set (20/50 for the ...
3
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1answer
68 views

Determine number of values less than mean in one pass through list

The problem statement is as follows: Can we determine precisely the number of elements less than the mean in a list $A$ of $n$ numbers in only one pass through the array (starting at $A_1$ and ...
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1answer
37 views

Singleton in a simple SBM

I can't work out the solution to the following exercise: We have $2n$ vertices grouped in $2$ clusters of equal size. The probability of having an edge between $i$ and $j$ is $p$ if $i$ and $j$ are ...