# 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.

167 questions
Filter by
Sorted by
Tagged with
50 views

### Does randomness help depth?

Suppose we have a $RNC^i$ or $BPNC^i$ algorithm for a problem, is it suspected that the problem has an $NC^i$ algorithm or just an $NC^j$ algorithm for some $j\geq i$? Is there any evidence for ...
• 13k
140 views

### Parity of the sum of pseudorandom bits over a non-sparse set of inputs

Suppose I have a pseudorandom function (in the theoretical sense) $X\colon\{0,1\}^{n+m}\rightarrow\{0,1\}$ (where $m$ is polynomial in $n$) and a non-empty set $S\subseteq\{0,1\}^m$ ($S$ is not sparse,...
• 369
373 views

• 828
419 views

### 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
299 views

### 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
251 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
136 views

### 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 ...
• 13k
122 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
116 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
302 views

### 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 ...
• 14k
235 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 "...
• 4,471
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
233 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 ...
• 6,852
1 vote
321 views

### 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 ...
386 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?
• 13k
152 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....
• 5,656
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 ...
529 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 + ...
• 4,471
125 views

• 2,077
342 views

### 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 ...
• 13k
587 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 ...
### 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\...