16
votes
Suppose $\mathbf{P} = \mathbf{BQP}$. Then what is randomness? Would it even exist at all?
P and BQP are decision-problem classes, i.e. the correct output is always a deterministic functions of the inputs. The only question is whether randomness helps "along the way" to speed up computing ...
- 7,175
13
votes
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
12
votes
Suppose $\mathbf{P} = \mathbf{BQP}$. Then what is randomness? Would it even exist at all?
The questions touches on some very interesting issues regarding quantum computation (and randomness). BQP is the class of decision problems that can be solved efficiently (in polynomial time) but it ...
- 5,983
12
votes
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 ...
- 9,545
10
votes
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 ...
- 14.2k
9
votes
Accepted
Graph that maximizes minimum hitting time?
It is well known that a barbell graph (two cliques of size $n/3$ connected by a path of length $n/3$) has average hitting time $\Omega(n^3)$, but I believe the same applies to minimum hitting time (...
- 50.6k
9
votes
Accepted
When does randomization stops helping within PSPACE
There is a difficulty with the premise of your question — "when does randomization stops helping within $\mathrm{PSPACE}$ — because it suggests that the computational classes $\mathrm{X}$ ...
- 8,801
9
votes
Accepted
Does two-sided error have more capability than one-sided error?
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 $\...
- 800
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. ...
- 1,657
8
votes
Accepted
Example of pairwise independent random process with expected max load $\sqrt{n}$
Here's how to do it.
First, choose a random $k$ between 1 and $n$ to be the "crowded bin".
Next, choose a random permutation $\pi$ of $1,2,\ldots, n-1$.
Now, for $1 \leq i \leq n-1$,
$$ \mbox{put ...
- 23.9k
8
votes
Randomized Polynomial Hierarchy?
Clearly, $\mathrm{RPH}\subseteq\mathrm{BPP}$. On the other hand, $\mathrm{BPP}=\mathrm{ZPP^{promiseRP}}$ (Buhrman&Fortnow, pdf), so the only way the hierarchy didn’t collapse to (at most) the ...
- 15.6k
8
votes
Evidence that there is some problem in BQP distinct from BPP?
Scott Aaronson has been addressing this topic: http://arxiv.org/abs/0910.4698
Related hardness results:
Boson sampling: http://arxiv.org/abs/1011.3245
Commuting circuits: http://arxiv.org/abs/1005....
- 5,436
8
votes
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 ...
- 26.7k
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 ...
- 11.3k
6
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 ...
- 14.2k
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 ...
- 11.3k
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).
- 544
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
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 ...
- 441
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 ...
- 11.3k
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 $\pi \in Aut(f)$, ...
- 36.4k
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 ...
- 13.9k
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 ...
- 10.2k
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/...
- 23.9k
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 ...
- 2,059
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 $\sigma^2=2\sum_i\lambda_i^{-2}$....
- 898
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 ...
- 1,733
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