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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PP and the most significant bit of functions in #P
I've found the following sentence (and some variants) in a lot of places, namely in Arora and Barak's Computational Complexity: A Modern Approach.
Intuitively, PP corresponds to computing the most ...
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How can $R_HL$ differ from $RL$?
https://complexityzoo.net/Complexity_Zoo:R
RL: Randomized Logarithmic-Space Has the same relation to L as RP does
to P. The randomized machine must halt with probability 1 on any
input. It must also ...
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Algorithm to select a random bit string with constraints
Problem Description
Given $a, b, n \in \mathbb{N}$ with $a < b < n$.
Let $M$ be the set of all possible bit strings of length $n$ which begin and end with one and have at least $a$ and at most $...
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Probabilty of Elements being smaller than a specific value
Right now i am looking at the following statement, but i cant grasp why it is correct.
Can somebody help?
"If we look at F0 uniformly distributed (and, say, pairwise independent) elements of
[0, ...
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Combine Las Vegas and Montecarlo probabilistic algorithms to improve chance of finding correct answer
Let's say that I have a Las Vegas algorithm for a given problem (whose answer is true/false for simplicity) with a chance of ...
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Error correcting codes for transmitting a real value $x$ in [0,1], minimizing the reconstructed distance to the original $x$
I have been thinking a bit about error correcting codes, in particular the following problem:
Consider the problem of transmitting a single real number $x \in [0,1]$ over a lossy connection, where ...
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Probability of this random selection
Suppose we have an array of $n$ integers. Suppose that we pick one of these elements uniformly at random and call it $x$. Suppose that $\log n$ elements are also sampled (uniformly at random) from the ...
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Probabilistic methods for undecidable problem
An undecidable problem is a decision problem proven to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. I wonder if there are examples of probabilistic ...
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Time complexity analysis for Searching in a Hash table
I want to analyse the time complexity for Unsuccesful search using probabilistic method in a Hash table where collisions are resolved by chaining through a doubly linked list. And the doubly linked ...
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Probability that an Algorithm Deviates from Its Behaviour after Multiple Rewindings
I do have a seemingly fundamental question that I somehow struggle to intuitively make sense of.
Setting:
Let us consider a randomized algorithm $R$ that has $t$ steps. In each step, it is fed with ...
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Algorithm for allocating resources; one resource per one user who accepts it
I am looking for an algorithm for the following problem:
I have a set of users and a set of books.
Every user has their own set of favorite books, which may be empty, and is a subset the set of books.
...
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Semi-bounded probabilistic polynomial-time, is it equal to BPP?
The complexity class $\mathsf{BPP}$ is typically defined as the class of all problems for which:
Running an algorithm once takes polynomial time at most.
The answer is correct with the probability at ...
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Rank of random binary string with Bernoulli distribution
For $1\ge p_1 \ge \dots \ge p_n \ge 0$, and for $i\in[n]$ draw $k$ iid binary strings with $m$ length:
$$X_{i,1},\dots,X_{i,k}\stackrel{iid}{\sim} \text{Bernoulli}(p_i)^m.$$
Viewing these binary ...
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Lower bound for ϵ-tester with one-sided error for the "2-injective" property of functions
An $\epsilon$-tester given an input and a property, is defined as follows:
If the input holds the property then the tester should accept with probability at least $\frac 2 3$. Otherwise if the input ...
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Weighted sample of ~k elements from array in O(n) time?
I have an array $a$ with $n$ elements, all of which have an associated weight. For example:
$a = \{ (A,2), (B,5), (C,9), ..., (Z,1) \}$, such that element $A$ has weight $w_A=2$, element $B$ has ...
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What are practical applications for markovs algorithm(string rewriting systems)
:)
Currently I am browsing through the internet on the hunt for a topic for my bachelors thesis. Whilst being on Reddit I discovered an interesting repo (https://github.com/mxgmn/MarkovJunior) for a ...
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Usage cases of AMS algorithm
This question is about the second frequency algorithm described in N. Alon, T. Matias, and M. Szegedy The space complexity of approximating the frequency moments. Specifically, I am asking about the ...
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Frequency moments
I have a question about the second algorithm of N. Alon, T. Matias, and M. Szegedy. The space complexity of approximating the frequency moments that specifically concerns computations of the second ...
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Learner Algorithm Time & Sample Complexity
Let $X=R^{2}$. Let $u=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right),\
w=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right),\ v=\left(0,1\right)$
and $C=H=\left\{h\left(r\right)=\left\{\left(x_{1},x_{2\ }\...
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RP with very small error = P
I was asked to show the equality $ RP(1 − 2^{-2^{n}}) = P $, which seems wrong to me (?).
The $ \supseteq $ direction is obvious, and I want to show the other direction.
My first intuition was to run ...
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Verify if array is orthogonal
Orthogonal arrays often appear in probabilistic algorithms. They can be efficiently constructed from, e.g., BCH codes.
But is there an efficient algorithm that can verify if an array is orthogonal? I ...
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Does the reliability of polynomial hashing depend on whether the modulus is prime, for coprime base and modulus?
A polynomial hash of a string $s$ with base $b$ and modulus $M$ is defined as
$$
H(s) = (s_0 + s_1 b + s_2 b^2 + \dots + s_{n-1} b^{n-1}) \mod M.
$$
I have proven (and this is quite obvious) that ...
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Are turing machines & equivalents with infinite sized random programs still turing machines?
Are turing machines with an infinite program tape that is completely random, or another example is a Game of Life simulation on an infinite randomly initialized grid, still turing machines, so to ...
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Why does random noise in recurring task periods result in uniform period offsets?
I have a recurring task which finished just now. I schedule it to run every ten minutes; the task will reoccur $10n$ minutes from now for all positive $n$. If instead I choose 50/50 between ten ...
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Given only the expected runtime of an algorithm, what can Markov's inequality tell us about its worst-case runtime?
The following is exercise 3.8 from the first edition of Mitzenmacher and Upfal's Probability and Computing: Randomized Algorithms and Probabilistic Analysis.
Suppose that we have an algorithm that ...
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Question about "with high probability"
An event that occurs with high probability is one whose probability depends on a certain number $n$ and goes to $1$ as $n$ goes to infinity, i.e. it can be made as close as desired to $1$ by making $n$...
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Distribution independence property testing
I have been reading the proof in the following paper, and I am unable to understand some parts in the proof. This paper shows that a distribution $A$ over $[n]\times[m]$, $n\geq m$, can be $\epsilon$-...
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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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Computing a threshold function
Let $f$ be any function from $\{0, 1\}^{n}$ to $\{-1, 1\}$. For a given $f$, let us define another function $g_f$ as
\begin{equation}
g_f(x) = \sum_{x \in \{0, 1\}^{n}} f(x).
\end{equation}
Let us be ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
