# 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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• 818
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### Chernoff bound for weighted sums of Bernoulli random variables

I tried to prove the following Chernoff-type bound when doing research, and found that it is of indepent interest. Let $X_1, \dots, X_n$ be independent random varibles such that each $X_i$ is a ...
2k views

### Expected number of random comparisons needed to sort a list

Consider the task of sorting a list x of size n by repeatedly querying an oracle. The oracle draws, without replacement, a random pair of indices (i, j), with i != j, and returns (i, j) if x[i] < x[...
• 429
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### Hardness assumption: an NP-complete problem whose ratio of hard instances do not tend to zero?

I am wondering about the following property $\text{(P)}$ of an $NP$-complete language $L$ \begin{align}\exists M\text{ a polytime machine}\lim_{n\to\infty}P(\text{M solves a random instance of size...
• 153
1 vote
249 views

### Converse form of Chernoff bound

Suppose $X_1,\ldots,X_n$ are i.i.d. random variables, all with mean $p$, and we have the statement that $\Pr\left[\sum_i X_i\geq 0.8n\right]\geq 0.9$, can we use this to conclude anything about $p$ (...
• 251
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### Derandomizing arbitrary width *read-many* and *ordered* branching programs?

Modifying following TedP We know that derandomizing width $5\leq k\in O(1)$ read many branching programs is equivalent to $BPNC^1=NC^1$ and derandomizing width $k\in\Omega(n)$ read once ordered ...
• 12.9k
121 views

### Creating a random tree (BST) on $n$ elements using a random sequence of zeroes and ones

We have a sorted list of $n$ numbers and we shall create a BST for these numbers. We create a random sequence of zeroes and ones of length $n$. We shall make use of this random binary sequence to form ...
• 137
1 vote
112 views

### Random variates generation in discrete-event simulation models

In discrete-event simulation, most university textbooks (e.g., Law & Kelton, Banks etc.) state that for generating variates for each random variable (e.g., interarrival time, service time etc.) in ...
• 21
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### Provable BPP Hierarchy

No Time Hierarchy theorem is known for $\mathsf{BPTIME}$, however, consider the following simple modification of the definition: A language is in $\mathsf{ProvableBPTIME}[f(n)]$ if there is a ...
• 14.1k
220 views

### Converting a Bernoulli to a Gaussian

It is not hard to see that, given one sample from a univariate unit-variance Gaussian $X\sim \mathcal{N}(\mu,1)$ with unknown $\mu$ s.t. $0<|\mu|\leq 1$, one can simulate one draw from a "...
• 4,461
201 views

### Generating $k$ random bits from a pdf with entropy $H(p) = k$

All the sources online say that, intuitively, a distribution with entropy $k$ has $k$ bits of pure randomness in it. So can we formalize this as follows? Suppose I can only sample from my distribution,...
• 607
220 views

### Upper bound on Independence Number of Random Regular Graph with degree $\Theta(\sqrt{|V|} \log^2 |V|)$

Let $G=(V,E)$ be a random $\Delta$-regular graph with $\Delta \in \Theta(\sqrt{|V|} \log^2 |V|)$. I'm analysing an algorithm having asymptotic running time crucially depending on the Independence ...
• 6,842
1 vote
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### How come Wikipedia says that Random Turing Machines can provide uncomputable output?

Wikipedia article mentioned : Hypercomputation The third paragraph starts off with: Technically, the output of a random Turing machine is uncomputable; however, most hypercomputing literature focuses ...
378 views

### Is there an assumption that implies $P=ZPP$ which is not known to imply $P=BPP$?

There are assumptions that are known to imply that $P = BPP$. For example, if there exists a function in $E = DTIME(2^{O(n)})$ that has circuit complexity $2^{\Omega(n)}$, then $P = BPP$ [1]. Clearly, ...
76 views

### Randomized Reduction for Maximization Problem

I have two maximization problems $P_1$ and $P_2$ where the decision version $L_1 = \{(x, t) : \operatorname{Val}_1(x)\ge t\}$ of $P_1$ is $\mathsf{NP}$-complete. Let $f:P_1\to P_2$ be a randomized ...
• 101
246 views

### Does two-sided error have more capability than one-sided error?

From $P=RP$ extrapolation we might think $EXP=REXP$. What evidence do we have $BPP\subseteq REXP$? What consequence $REXP\subseteq BPP$ gives other than what $EXP\subseteq BPP$ gives?
• 12.9k
146 views

### BPP fragment of a PSPACE complete problem

Consider a PSPACE-complete problem (e.g., TQBF). Is there a sub-problem in BPP, that is not known to be in P? Is there a general technique of finding such sub-problems? Are any of them "natural" (i.e....
• 5,646
87 views

### Expected value of a random experiment in a graph

I need to find the expected value of R in the random experiment below. $$R = \frac{1}{K} \sum_{C \in \mathcal{H} } \ [\frac{1}{2} |V(C)| * (|V(C)| - 1) - |C|]$$ $\mathcal{H}$ is a partition on ...
522 views

### Deterministic error reduction, state-of-the-art?

Assume one has a randomized (BPP) algorithm $A$ using $r$ bits of randomness. Natural ways to amplify its probability of success to $1-\delta$, for any chosen $\delta>0$, are Independent runs + ...
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• 2,077
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### UnambiguousSAT reductions

Let $\Pi$ be an $\mathsf{NP}$-complete problem. It is standard that $3SAT$ and $\Pi$ are reducible from each other. Let UnambiguousSAT, or USAT for short, denote the promise problem which is 3SAT but ...
• 12.9k
563 views

### Connectivity of a random regular graph of degree $d$

An Erdős–Rényi graph over $n$ vertices and average degree $d$ is not connected w.h.p. iff $d < \log n$. I was wondering for what the degree $d$ would a random regular graph of degree $d$ be ...
459 views

### Can true randomness (provably) be replaced with Kolmogorov randomness for RP?

Have there been any attempts to show that Kolmogorov randomness would be sufficient for RP? Would the probability used in the statement "If the correct answer is YES, then it (the probabilistic Turing ...
• 3,043
187 views

### Probability of random variable $X$ less than $max(Y_i)$

For $\lambda > \mu$, consider independent random variables (i) $X \sim Pois(\mu)$ and (ii) $Y_i \sim Pois(\lambda),\;i\in{1,2}$. Question : Can we upper-bound the following? \mathbb{P}\big(X\...
242 views

### The average number of compressible strings in a random set of random strings

In the book Elements of Information Theory (p.446), it is stated: ...although there are some simple sequences, most sequences do not have simple descriptions. Similarly, most integers are not simple. ...
• 133
Let $G \sim G_{n,p}$ be a binomial random graph and consider a sequence of graphs $\{H_n\}_n$. When $|H_n|=\mathcal{O}(1)$, then one knows the exact threshold $p_0$ for embeddabiliy of $H_n$ in $G$. ...