# Questions tagged [randomness]

Randomness is a way to mathematically model uncertainty. We often assume to have access to some well-defined source of random numbers, or that input values or events follow some probability distribution.

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### Under which conditions a given program is deterministic on x86_64 machines?

Given a certain x86_64 "vanilla" binary, without micro-architecture instructions, which can therefore be executed by any x86_64 computer, what are the conditions for the result to be ...
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### Benford's law is not working?

So I recently came across Benford's law and immediately tried to code it out but the answer I got was rather confusing. I've been thinking about it for about an hour now and still can't get it, I ...
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### How does random number probabilities work in distributed systems?

Let's say I have a code which does coin toss at 25% win probability. If I run this code n number of times in the same system it will yield ~25% win. If I run this code on n devices (once each) will ...
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### Sampling from specific random distribution on sets

I have a random distribution on sets in mind, that has three parameters: $n, w, k$. The goal is to sample sets of $k$ integers from $[0, n)$ (without replacement) such that the elements within each ...
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### Permutation with stable partitions

I am searching a fast pseudo-random permutation function with the following requirements. Given a predefined set of values $V$ and an integer $k$. Split $V$ to $k$ subset, and iterate over any subset ...
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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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### 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: ...
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### 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 ...
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### How to generate a uniform random sample of unique vertex pairings from a undirected graph under constraint?

I'm working on a research project where I have to pair up entities together and analyze outcomes. Normally, without constraints on how the entities can be paired, I could easily select one random ...
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### Problem with proving that $RP \subseteq NP$ : a non-deterministic TM for a language $L \in RP$

I'm having a small issue with wikipedia's proof that $RP \subseteq NP$: An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic ...
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### Is the BPP class closed for union and intersection?

Just like the title says. I want to prove that given two languages $L_1,L_2 \in BPP$, $L_1 \cup L_2 \in BPP$ and $L_1 \cap L_2 \in BPP$
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### Reference asking: phase transition in SAT

This is not a technical question, I hope this community has a room for such questions, but I will delete it in case this is inappropriate. It has been experimentally observed (e.g. here) that when ...
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### PAC learning vs. learning on uniform distribution

The class of function $\mathcal{F}$ is PAC-learnable if there exists an algorithm $A$ such that for any distribution $D$, any unknown function $f$ and any $\epsilon, \delta$ it holds that there exists ...
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### How to implement random sampling with continuous variables?

How functions like rnorm in R (and similar functions) create a random sample ? If I want to implement one algorithm to simulate this procedure what can I do? When you have the pdf or pmf of a ...
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### Shuffling a collection too large for memory

I have a set that is too large to hold in memory but I have a function that I can use to compute a value within the set given an index. I'm curious if there's a standard way to shuffle this set, a ...
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### Algorithm For Random Distribution of Prizes Over A Given Time Period

I have been assigned a task to come up with an algorithm for evenly distributing prizes over a given time frame, I am looking for advice as I am a bit stumped on where to start. Here are my ...
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### Grid Puzzle Split Algorithm

I want to generate a random partition of an $N\times N$ grid into $N$ connected groups having $N$ tiles each. How would I do this? Max grid size will be 10x10. Below is an example for a 5x5 grid.
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### Is there a theorem which relates calculating the total number of a combinatorial object with picking one at random?

A common algorithmic challenge is to generate an object of a certain kind, uniformly at random. For example, generating a random permutation of size $k$ from a given (multi)set of $N$ characters, as ...
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### generalizing ball-bin problem to k-universal family

I am trying to solve a question in the book on Probability and Computing by Michael Mitzenmacher, Eli Upfal. The question asks to generalize ball-bin problem for 2-universal hashing to $k$-universal ...
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### How likely is for a pseudorandom number generator to generate a long sequence of similar numbers?

How likely is for a pseudorandom number generator to generate a long sequence of similar numbers? "Similar numbers" could be same numbers, or numbers from a given range. For example, if we consider ...
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### Can neural network process randomness?

So the question is : Is it theoretically possible to feed a neural network with some random values to expect an output since randomness is a lack of knowledge in most case. For this question, I've ...
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### Reservoir Sampling vs Round Robin

You are given a List of numbers (length unknown). Let's say the length is 10. GetRandom(List) is called once. If implemented correctly, each number has 1/10 probability of being returned. GetRandom(...
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### If current time in milliseconds is considered good enough random seed for a pseudorandom number generator, why not just use that time directly?

I was reading about pseudorandom number generators and how they need a seed and how that seed is usually current system time in milliseconds. One of the most common algorithms is the linear ...
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### Simple generator of pseudo-random permutations of variable length short sequence

The problem in front of me is to write a function (from scratch) to permute n elements, where n is an argument. I decided to break it down to applying Knuth's shuffles algorithm, therefore I needed to ...
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### Need help regarding random password generator: Urgent [closed]

I had generated a random password from a website which unfortunately has put me into trouble. The password contains 26 characters which includes alphabets, numbers and special characters. My trouble ...
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### Finding efficient randomized algorithm

I'm doing a course on randomized algorithms and I've encountered a question that I'm struggling to solve. Given a system of $m$ linear equations with $n$ variables over finite field $\mathbb{F_2}$ ...
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### Derandomize MAX-CUT problem using $\log n$ bits

Consider the MAX-CUT problem. We can flip $n$ coins to generate a random cut, and by linearity of expectation we get that with "good probability" our cut we'll be bigger then $\frac{n}{2}$. Using ...
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### Can we generate random numbers using irrational numbers like π and e?

Irrational numbers like $\pi$, $e$ and $\sqrt{2}$ have a unique and non-repeating sequence after the decimal point. If we extract the $n$-th digit from such numbers (where $n$ is the number of times ...
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### What doest it mean: “computer is an intelligence amplifier”?

There is one example in Kolmogorov complexity books and related articles: Consider we have a monkey at a typewriter and a monkey at a computer keyboard. If the monkey types at random on a typewriter,...
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### Simulating a probability of 1 of 2^N with less than N random bits

Say I need to simulate the following discrete distribution: $$P(X = k) = \begin{cases} \frac{1}{2^N}, & \text{if k = 1} \\ 1 - \frac{1}{2^N}, & \text{if k = 0} \end{cases}$$ The most ...
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### How good (or bad) is my makeshift PRNG?

Say I have designed a makeshift PRNG for my personal amusement, now I would like to see how good it is. How do I benchmark its "randomness"? Ideally, I want to know a statistics test, such that if I ...
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### Random observations of a total ordering, how much they tell us?

Suppose we have a total ordering over elements $a_1,a_2, ..., a_n$, meaning there is permutation $\pi$ such that $a_{\pi(1)}<a_{\pi(2)}<...<a_{\pi(n)}$. But we don't know $\pi$. What we know ...
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### Generating uniformly random bits from a stream of arbitrarily biased bits

Say we have a function called GenBiasedBit. This function returns 1 with probability p (where p is an unknown real number between 0 and 1 exclusive) and returns 0 with probability 1 − p. How could I ...
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### Is von Neumann's randomness in sin quote no longer applicable?

Some chap said the following: Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin. That's always taken to mean that you can't generate ...