# Questions tagged [randomness]

Randomness is a key component of probabilistic algorithms, many combinatorial aarguments, the analysis of hashing functions, and in cryptography, among other applications.

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### Finding a random regular graph with degree d

I'm trying to find undirected random graphs $G(V,E)$ with $|V|$ = $d^2$ for $d \in \mathbb{N}$ such that $\forall v \in V: deg(v) = d$. For $d \in 2\mathbb{N} +1$ this trivially is impossible as no ...
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### How many neighbors does a vertex has which are closest to a source vertex in random regular graphs?

Let $G=(V,E)$ be an undirected, random $r$-regular graph. Let $s,t\in V$, and denote by $N(v)=\{u\in V\mid (u,v)\in E\}$ the neighborhood of $v$. I'm looking for the distribution of the number of ...
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### Suppose $\mathbf{P} = \mathbf{BQP}$. Then what is randomness? Would it even exist at all?

DISCLAIMERI do apologize in advance if this question turns out to be silly, for some trivial reason that I may be overlooking in this moment. Suppose for a moment that $\mathbf{P} = \mathbf{BQP}$ ...
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### When does randomization stops helping within PSPACE

It is known that adding bounded-error randomization to PSPACE doesn't add power. That is, BPPSAPCE=PSPACE. It is famously unknown whether P=BPP, but it is known that $BPP\subseteq \Sigma_2\cap \Pi_2$....
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### Example of pairwise independent random process with expected max load $\sqrt{n}$

This question was previously posted at https://math.stackexchange.com/questions/1220292/example-of-pairwise-independent-random-process-with-expected-max-load-sqrtn where it has no answers. I now ...
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### Satisfiability threshold and partially random formulas

My understanding is that if we have a totally random k-SAT formula, for a ratio $m < \alpha n$ far enough below the satisfiability threshold, we can solve for satisfiability in polynomial time (...
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### Deterministic Randomness Extractors

I have read in several papers it is well known that deterministically extracting even one bit from a weak source is impossible. Could someone explain why?
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### Randomly Discovered Algorithm/Counterexample

I was reading Scott Aaronson's blog, and one of the comments sparked a question. "if P!=NP, this would be a general, conceptual result, so you’d expect the proof to be explanatory and in particular ...
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### Natural theorems proven only “to high probability”?

There are plenty of situations where a randomized "proof" is much easier than a deterministic proof, the canonical example being polynomial identity testing. Question: Are there any natural ...
127 views

### Largest size for randomness extractor

Suppose we have a source $X$ with min-entropy $\ell$. A randomness extractor is defined as a function $f$ which satisfies the total variation $||f(X, R)-U_M||_{TV}\leq \epsilon$ where $R$ is an ...
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### Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
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### Sort with random deviations

I'm looking for an algorithm that will take a sorted array of numbers and generate a random permutation of this array in such a way that the probability of finding a larger element earlier in a ...
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### Size complexity of probabilistic two-way automata for a Boolean function

I'm interested in computing Boolean functions $f:\{0,1\}^n\rightarrow\{0,1\}$ with two-way finite automata and I will measure the complexity of a Boolean function by the number of states for the ...
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### Will quantum computing pave the way for native, true RNGs?

Obviously, regular computers can't generate random numbers on their own, since they're inherently systematic machines. Would quantum computing be able to run a true RNG without a seed based off user ...
Consider two sequences $u_1 \geq u_2 \geq ... \geq u_n$ and $l_1 \geq l_2 \geq ... \geq l_n$ with $u_i \geq l_i$ for every $i$. Let $\mathcal{G}(l_{1:n},u_{1:n})$ be all undirected unweighted simple ...