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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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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Proving $BPP[\frac{1}{3},\frac{2}{3}] = BPP[a,a+\gamma]$ where $0<a<a+\gamma<1$
I can prove $BPP[\frac{1}{3},\frac{2}{3}] \subseteq BPP[a,a+\gamma]$ by sampling the promised $BPP[\frac{1}{3},\frac{2}{3}]$ Turing machine $k$ times and accept if $k\cdot(a+\frac{1}{2}\gamma)$ of the ...
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Are real computers capable of producing true random series with a program of finite length?
Suppose that one has a Blum–Shub–Smale machine. Is it possible to write down a program of finite length that produces a series of numbers, with each consisting of a finite number of digits, such that ...
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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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2answers
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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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1answer
33 views
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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1answer
50 views
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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1answer
36 views
Uniformly random iteration of subtrees
I want to iterate through the first $k$ elements of a randomly ordered list containing all subtrees for a given tree. The definition of subtrees that I'm using is: "A subtree of $T$ is a subgraph of $...
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1answer
43 views
PRNG for a gaussian distribution?
I need suggestions for algorithm for Pseudo random number generators that will produce a near gaussian distribution (bell curve) for automatically generating test data. I know that PRNG produce ...
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1answer
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Sublinear Homomorphism Property Testing Counter Example
This is a homework question, so I'm not looking for answers, just general guidance.
I'm looking at a Sublinear Algorithms survey where (Group) Homomorphism property testing is discussed. The case of ...
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1answer
35 views
Concentration bound for sum of dependent geometric random variable?
consider following persudocode:
i=0
while(i< k):
uniformly pick u,v in V
if(uv in E):
remove uv form E;
i++;
let $T$ be the number of ...
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0answers
34 views
Matrix with zero spectral radius
I need an algorithm which generates a random matrix with spectral radius equal zero.
The only solution I have so far is to generate two vectors $v,w$, normal onto each other ($v\perp w$), and then ...
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1answer
73 views
Is there a name for this shuffle algorithm?
Is there a standard name for the following algorithm? At each step, the code picks an index, remove the object at that index, and remember to never pick that index again. See the following ...
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2answers
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Random conditional association
I have a really specific problem to solve and i'm looking for an existing algorithm to help me (i hope i'm in the right section of stackExchange).
I would have to make "random" association between ...
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1answer
27 views
On uniform randomness of the weight of the remaining edges of a graph after deleting some of them
Suppose we have a graph $G(V,E,W)$, where $V$ and $E$ are the set of vertices and edges and $W$ is non-negative weight on the edges. Let $w(e)$ be the weight of edge $e$ and $N(e)$ be the neighboring ...
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If P=BPP, then Is it correct that IP=NP?
I'm now studying Interactive Proof System in Goldreich's textbook and here. I have the following question:
Suppose P=BPP, then this show that randomness doesn't give us power over deterministic ...
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1answer
144 views
Is it possible to simulate a fair coin with a finite number of tossing of a biased one?
It is a classic problem to simulate a fair coin with a biased one.
According to Fair Coin (wiki),
John von Neumann gave the following procedure:
Toss the coin twice.
If the results ...
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2answers
175 views
How to determine seed collision probability in a PRNG?
I want to use a PRNG to generate random patterns. I would provide the PRNG with a hash value as a seed. Ideally, the seed size would be 64-bit or 128-bit and I would expect no collisions if the seeds ...
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1answer
27 views
Is there a complexity class “BQP without error”?
I was wondering if there is a complexity class for problems that can be solved efficiently by a quantum computer such that it always gives the right answer? For example the Deutsch-Josza algorithm ...
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Generate higher dimensional pink noise
1D Pink noise, is easy enough to generate. See https://www.dsprelated.com/showarticle/908.php for example.
What about higher-dimensional pink noise, such as 2D or 3D pink noise? Is there an algorithm ...
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1answer
102 views
How could you characterize “true randomness” of a finite sequence?
This question occurred to me while reading http://arxiv.org/abs/1806.08762/ Any observed sequence is necessarily finite, and any finite sequence is computable, either by explicitly storing all the ...
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1answer
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Finding the Entropy of a random experiment with probability of $\frac{1}{3}$
Entropy is the randomness collected by an operating system or application for use in Cryptography or other uses that require random data. The formula for Entropy is $$H(p_1, ..., p_k)=-\sum_{i=1}^{k} ...
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1answer
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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 ...
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1answer
54 views
Are there any stochastic models of non determinism in the rate of program execution?
We know that typical operating systems and high level languages (especially those with garbage collection) cannot be used for real time operating systems. Java & jet engines don't mix, and ...
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2answers
57 views
Would using mouse position in the below scenario generate true random numbers?
I'm having trouble wrapping my head around random number generation so I figure it would be better to ask now, rather than screw up later.
Let's say I want to write a function that generates a random ...
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3answers
2k views
Problem with the pseudo random number generator One-Time-Pad
I've started learning cryptography in class and we've come across One-Time-Pads, in which the key (uniformally agreed upon) is as long as the message itself. Then you turn the message into bits, do $...
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1answer
80 views
Placing small circles randomly inside a larger circle, where no two small circles intersect
Sorry if this is the wrong place to ask, was unsure...let me know if it belongs some where else.
So i am trying to work out a way to write an algorithm to place a series of small circles of a set ...
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1answer
80 views
Why doesn't this put $BPP$ in $NP$?
From Sipser Gacs we know $x\in L(M)$ for a machine $M\in BPP$ $\iff$ $$\exists t_1,\dots,t_{|r|}\forall r\in\{0,1\}^{|r|}\vee_{i\in\{1,\dots,|r|\}}M(x,r\oplus t_i)=1.$$
From Adleman we know $x\in L(M)...
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1answer
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RNG: removing bias by counting bits
I have a hardware RNG that sends me some random bits. Due to nature of that RNG, the source is biased. Then, the bitstream is processed in a following way to remove the bias:
two counters are used, ...
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1answer
151 views
Expected Linear time Minimum Spanning Tree algorithm
I am trying to understand the proposed "Randomized Linear-Time Algorithm to Find MST".
My findings:
I have read and search almost every available resource( main paper, wiki, reports on paper, lecture ...
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1answer
38 views
Looping through random integers - will it halt with probability 1?
Say I have a simple program that has the pseudocode like this:
...
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1answer
36 views
Randomly choose a line - algorithm
We have a large file that can't fit into internal memory. How do we randomly pick one line so that each line has the same probability to be picked?
And how do we randomly pick such n lines so that ...
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3answers
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How to select a binary tree node uniformly at random
The exercise I'm trying to solve is
You are implementing a binary search tree class from scratch, which, in addition, to insert, find and delete, has a method ...
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2answers
51 views
Indexing a random permutation
I am curious if there exists a method for specifying a permutation $F_k: X \to X$ with a small(ish) $k$.
Something that comes very close to my goal is a block cipher, say AES. But block ciphers have ...
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2answers
319 views
What Good Is Kolmogorov Complexity Since It Is Relative?
Kolmogorov complexity is relative to a choice of Universal Turing Machine. Because of the Invariance Theorem, the difference in complexity assigned by two Universal Turing Machines is bounded by a ...
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1answer
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Fitness model for scale free networks
In order to generate scale-free networks, we can use this algorithm, derived from Barabási–Albert model:
1) we assign every node a "weight" $\theta_i$ (or two in the direct case).
2) we place $m$ ...
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3answers
83 views
Given a sufficiently powerful computer and model, can dice rolls be predicted?
I guess that this is a chaos, randomness and modelling question. Can it be conceived that a sufficiently powerful (perhaps quantum) computer might be able to accurately predict the scores on a ...
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0answers
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What is the relationship between entropy rate and quantization?
I have a totally random source of signal data that looks like a typical normal distribution. I've included an image as I like pictures:-
The source has a mean of 0, and a standard deviation of 1. ...
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1answer
33 views
TRNG by hamming distance of processor tics
I hardly remember on a lecture, where it was told, that the distance of the practical actual running-speed of a processor to the previous tic is used, to generated a true random entropy for seeding. ...
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2answers
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What makes a pseudorandom generator, a high quality one?
Reading this answer to this SO question: Why do we not combine random number generators?, it talks about
very high-quality PRNG (Pseudo Random Number Generator)
so it makes me wonder what ...
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1answer
653 views
How to understand the analysis of expected running time of randomized quick-sort in this paper?
I'm learning the book named Data Structures & Algorithms in Python.
On Page 557-558, there is a proof of the expected running time of randomized qucick-sort. I have some problems confusing me ...
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2answers
324 views
Shuffling a list while keeping order relative to related elements
I'm looking to shuffle a list of the elements $a_1,\dots, a_6, \dots, e_1, \dots, e_6$
while keeping two rules:
if I loop though the list and filter out a specific letter or number it should be in ...
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3answers
407 views
Why do we need Gibbs sampling (and MCMC)?
I just learned about Gibbs Sampling which is an MCMC method. Given a distribution $\pi$, we want to sample an item according to $\pi$.
Maybe my alternative suggestion would sound somewhat naive (even ...
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1answer
41 views
Difference in scale between mean page loading times and exponentially distributed page loading times?
Apologies if this is the wrong place to ask.
For my experiment, I have a single-page application and a multi-page application. I am testing them with an exponential distributed delay throttling. The ...
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2answers
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Algorithm to create tournament brackets
I'm designing a web app to host e-sports tournaments and want to create an algorithm that generates tournament brackets.
Given a list of participants' user ID's (minimum 4, maximum let's say 64), I ...
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3answers
2k views
Time complexity of this while loop
So, i would like to know the time complexity of the following codes:
...
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0answers
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Generating symmetric keys
Let's assume that we have to generate huge number of 256-bit symmetric keys. How to do it and how to be sure that it is safe. Perhaps it requires a strong mathematical backend. Don't hesitate to ...
