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Accepted

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

For constant $d \geq 3$, a random $d$-regular graph is connected with high probability. In fact, it is an expander with high probability. See for example this note by David Ellis. Friedman even showed ...
• 14.5k

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

I think the question being asked here is roughly "is there a sense in which we can replace the sequence of random bits in an algorithm with bits drawn deterministically from an appropriately long ...
• 548
Accepted

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

This answer gives exact formulas for the expected number of steps, with and without replacement. To be clear, we interpret OP's problem as detailed in OP's Python gist: each step of the process makes ...
• 10.8k
Accepted

### Theorem 2.4(i) in Valiant-Vazirani paper "NP is as easy as detecting unique solutions"

For notational convenience define r.v.s $T_S = \min\{i : |S_i| = 1\}$ (recalling $S_i = S \cap H_1 \cap \cdots \cap H_i$), and $T_H = \min\big\{i : H_1 \cap H_2 \cap \cdots \cap H_i = \{0^n\}\big\}$. ...
• 10.8k
Accepted

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

I think it is "easy" to come up with an assumption that implies one but not necessarily the other... (just write down a condition that is equivalent to P=ZPP)... however, a "natural" and non-uniform ...
• 27.5k
Accepted

This isn't my area, so many apologies if I say something incorrect: 1) "What evidence do we have that $\mathsf{BPP}\subseteq \mathsf{REXP}$?" Isn't this unconditionally true? It should follow from $\... 9 votes ### Expected number of random comparisons needed to sort a list I believe this takes$\Theta(n^2)$oracle calls without replacement, and$\Theta(n^2\log n)$oracle calls with replacement. Lower bound: Assume my list has a single pair$x[i], x[i+1]$out of order. ... • 1,685 7 votes Accepted ### Where does the "intuitive" understanding of Kolmogorov complexity fails The issue in play here is whether you use a self-terminating encoding (like your C example) or not. If you use a self-terminating encoding, then the subadditivity property does hold. If you don't (as ... • 14.5k 7 votes Accepted ### How come Wikipedia says that Random Turing Machines can provide uncomputable output? It's not uncommon for Wikipedia to say dubious things. Don't trust it as a primary reference. Beware that hypercomputation is potentially a "crank-adjacent" subject, so the Wikipedia ... • 12.3k 6 votes ### Is there an assumption that implies$P=ZPP$which is not known to imply$P=BPP$? If you are happy with impying$P=RP$(which implies$P = ZPP$) but not$P = BPP$, then there is the Stoquastic PCP conjecture (or its classical version, a SetCSP PCP conjecture). • 554 6 votes Accepted ### Sets of solutions which it is hard to uniformly sample from, but easy to integrate functions over? (Or compute expectations over?) There is no such problem. If it's hard to sample, it's hard to integrate. Here is a sketch of the reason why. Represent every solution$x$by a$n$-bit string$x_1,\dots,x_n$. If you can integrate ... • 12.3k 6 votes Accepted ### Converting a Bernoulli to a Gaussian Suppose you had such a randomized procedure that takes a value in$\{-1,1\}$and outputs a real number. Let$P$and$Q$be the output distribution on input$+1$and$-1$respectively. Consider the ... • 76 6 votes ### Expected number of random comparisons needed to sort a list Not a complete answer, but here's a start. Consider the index$p_{k} \in \{1,\dots n \}$to represent the$k$th object in the series, with respect to order. We have: $$x[p_{k}] < x[p_{j}] \... • 161 6 votes ### Are there any problems in \mathsf{BPP} that are known to be \mathsf{RP}-hard or \mathsf{coRP}-hard? It would be a very interesting result if one were able to present a language in BPP that is hard for RP (equivalently, for co-RP) under poly-time Turing reducibility (aka Cook reducibility). I'm ... • 2,261 5 votes ### Generating k random bits from a pdf with entropy H(p) = k The relevance of Shannon entropy is to repeated sampling: Given n independent samples from a source with binary Shannon Entropy k, you can extract nk(1+o(1) i.i.d. uniform bits as n tends to ... • 451 4 votes Accepted ### What is the probability that a random Boolean function has a trivial automorphism group? Yes. To your first question, the probability goes to zero double-exponentially fast. This can be calculated as follows. For each permutation \pi, we can bound the probability that \pi \in Aut(f), ... • 37.8k 4 votes Accepted ### Motivation for randomness extractors Here we have d \ll m, i.e., we start with a little bit of good randomness, and we end up with a lot. That's why it's called a "seed": you need something small to get you started, but you end up ... • 12.3k 4 votes Accepted ### Newman's lemma for distributional communication complexity There's nothing wrong in your proof, but you can do even better; by taking the average in$$ \mathbb{E}_r \mathbb{E}_{\mu} \mathbf{1}_{\Pi(x,y; r) \neq f(x,y)} \leq \varepsilon $$you can conclude ... • 14.1k 4 votes ### How to use a 𝑝-coin so a TM can decide an undecidable language in polynomial time? I'll take as given the existence of Chaitin's constant \Omega\in[0,1], and that knowing its first k bits is equivalent to be able to decide the halting problem for all Turning machines of size up ... • 10.6k 3 votes Accepted ### Deterministic error reduction, state-of-the-art? Doesn't van Melkebeek's lecture notes already give a O(1/\delta) bound? The bound there is \lambda at most O(\sqrt{\delta}) and we can get \lambda = O(1/\sqrt{d}) using existing constructions. ... • 96 3 votes ### Sum of Independent Exponential Random Variables For the Laplace distribution, if you use the Bernoulli bound you can write$$Ee^{u\sum_i X_i} = \prod_i \frac1{1-u^2/\lambda_i^2} \le \frac1{1-u^2\sigma^2/2},$$where \sigma^2=2\sum_i\lambda_i^{-2}.... • 958 3 votes ### The power of randomized logspace with two-way access to the random tape Well, here are a couple of observations. There's a famous PRG by Nisan that fools \mathsf{BPL}-type algorithms with seed length O(\log^2 n). Given two-way access to the seed, Nisan's PRG can be ... • 1,743 3 votes ### Password checking algorithm As much as you're being downvoted and attacked, your idea is absolutely right, correct, and valid. You've nearly reinvented bcrypt. Let's say we have encryption algorithm (doesn't matter which one): ... • 131 3 votes Accepted ### The average number of compressible strings in a random set of random strings The answer is less than 2^{n-k} of the sequences of length n have complexity less than <n-k. Due to the uniformity assumption on these sequences we just count. Consider all short programs of ... • 2,070 3 votes Accepted ### Distributions which are intractable to sample from? I'll expand my comment to an answer. Many combinatorial structures in graphs are actually NP-hard to sample from. The earliest example I can think of is JVV86 (Thm 5.1), which shows that there is no ... • 375 3 votes Accepted ### Weighted balls and bins Suppose you have n balls. Take a random ball from each non-empty bin. Your expected weight is approximately (1-1/e)T. This holds because the probability that a bin is empty is (1-1/n)^n \approx 1/... • 24.9k 2 votes ### Is true randomness and the physical Church-Turing thesis incompatible? The Church-Turing thesis is about (partial) functions \mathbb{N} \to \mathbb{N} (or \Sigma^* \to \Sigma^* for a finite alphabet \Sigma). How do you define a definite value based on some random ... • 3,053 2 votes Accepted ### Robustness to non-uniform randomness vs. one-sidedness Only one-sided deciders are robust to bad coins, under your definition; no other decider can be robust to bad coins. Let D be a decider that is not one-sided. For x \in \mathcal{Y}, let \... • 12.3k 2 votes Accepted ### Optimal bounds for k-wise non-uniform random bits$$s = \Theta( k \cdot ( t + \log n ) )$$As the question mentions, there is an upper bound of$s \le k\cdot\max\{t,\lceil \log_2 n \rceil\}\$ bits for the seed length. Specifically, sample a random ...
• 2,803

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