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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11 answers
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What is the most efficient way to generate a random permutation from probabilistic pairwise swaps?

The question I am interested in is related to generating random permutations. Given a probabilistic pairwise swap gate as the basic building block, what is the most efficient way to produce a ...
Joe Fitzsimons's user avatar
44 votes
7 answers
6k views

Truly random number generator: Turing computable?

I am seeking a definitive answer to whether or not generation of "truly random" numbers is Turing computable. I don't know how to phrase this precisely. This StackExchange question on "efficient ...
Joseph O'Rourke's user avatar
39 votes
9 answers
27k views

What is the difference between non-determinism and randomness?

I recently heard this - "A non-deterministic machine is not the same as a probabilistic machine. In crude terms, a non-deterministic machine is a probabilistic machine in which probabilities for ...
user avatar
37 votes
1 answer
1k views

Efficiently computable function as a counter-example to Sarnak's Mobius conjecture

Recently, Gil Kalai and Dick Lipton both wrote nice articles on an interesting conjecture proposed by Peter Sarnak, an expert in number theory and the Riemann Hypothesis. Conjecture. Let $\mu(k)$ ...
Hsien-Chih Chang 張顯之's user avatar
36 votes
1 answer
3k views

Consequences of $\mathsf{NP}$ containing $\mathsf{BPP}$

Many believe that $\mathsf{BPP} = \mathsf{P} \subseteq \mathsf{NP}$. However we only know that $\mathsf{BPP}$ is in the second level of polynomial hierarchy, i.e. $\mathsf{BPP}\subseteq \Sigma^ \...
Kaveh's user avatar
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32 votes
4 answers
4k views

Can a probabilistic Turing machine solve the halting problem?

A computer given an infinite stream of truly random bits is more powerful than a computer without one. The question is: is it powerful enough to solve the halting problem? That is, can a ...
Joey Adams's user avatar
31 votes
2 answers
1k views

Hierarchy for BPP vs derandomization

In one sentence: would the existence of a hierarchy for $\mathsf{BPTIME}$ imply any derandomization results? A related but vaguer question is: does the existence of a hierarchy for $\mathsf{BPTIME}$ ...
Sasho Nikolov's user avatar
25 votes
1 answer
777 views

Random self-avoiding lattice cycle within a given bounding box

In connection with the Slither Link puzzle, I've been wondering: Suppose that I have an $n\times n$ grid of square cells, and I want to find a simple cycle of grid edges, uniformly at random among all ...
David Eppstein's user avatar
23 votes
2 answers
2k views

Problem in BPP but not known to be in RP or co-RP

Is there an example of a natural problem that's in BPP but that's not known to be in RP or co-RP?
arnab's user avatar
  • 7,000
22 votes
2 answers
2k views

Bounds on $E[f(x)]$ in terms of $f(E[x])$ other than Jensen's inequality?

If $f$ is a convex function then Jensen's inequality states that $f(\textbf{E}[x]) \le \textbf{E}[f(x)]$, and mutatis mutandis when $f$ is concave. Clearly in the worst case you cannot upper bound $\...
Ian's user avatar
  • 2,727
22 votes
1 answer
487 views

Random functions of low degree as a real polynomial

Is there a (reasonable) way to sample a uniformly random boolean function $f:\{0,1\}^n \to \{0,1\}$ whose degree as a real polynomial is at most $d$? EDIT: Nisan and Szegedy have shown that a ...
Igor Shinkar's user avatar
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22 votes
1 answer
342 views

Belief propagation for approximate real 3LIN?

In a Science paper from 2002, Mezard, Parisi and Zecchina put forward the belief propagation heuristic for random 3SAT. Experiments indicate that the heuristic works well for ratios of constraints-per-...
Dana Moshkovitz's user avatar
21 votes
1 answer
335 views

A comparison of extractors in terms of tradeoffs between time, randomness and space ?

Is there a good survey that compares different extractors, concentrators and superconcentrators and lays out the best methods in terms of the tradeoff between randomness, time and space ?
Suresh Venkat's user avatar
20 votes
4 answers
706 views

Is there current research into the implemention of Randomness Extractors?

Has there been research into implementing randomness extractor constructions? It seems that extractor proofs make use of Big-Oh, leaving the possibility for large hidden constants, making ...
Phillip Mates's user avatar
20 votes
3 answers
1k views

Problems that are NP-complete under randomized or P/poly reductions.

In this question, we appear to have identified a natural problem that is NP-complete under randomized reductions, but possibly not under deterministic reductions (although this depends on which ...
Peter Shor 's user avatar
20 votes
2 answers
644 views

Bounded depth probability distributions

Two related questions about bounded depth computing: 1) Suppose that you start with n bits, and to start with bit i can be 0 or 1 with some probability p(i), independently. (If it makes the problem ...
Gil Kalai's user avatar
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20 votes
0 answers
812 views

Weighted Hamming distance

Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is not necessarily Theoretical Computer Science but I think similar things come up ...
Bjørn Kjos-Hanssen's user avatar
17 votes
3 answers
725 views

Randomize or Not?

This question is inspired by the Georgia Tech Algorithms and Randomness Center's t-shirt, which asks "Randomize or not?!" There are many examples where randomizing helps, especially when operating in ...
Lev Reyzin's user avatar
  • 12k
16 votes
1 answer
1k views

Number of Hamiltonian cycles on random graphs

We assume that $G\in G(n,p),p=\frac{\ln n +\ln \ln n +c(n)}{n}$. Then the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
Elely's user avatar
  • 309
15 votes
1 answer
898 views

Random monotone function

In Razborov-Rudich's Natural Proofs paper, page 6, in the part they discuss that there are "strong lowerbounds proofs against monotone circuit models" and how they fit into the picture, there are the ...
Kaveh's user avatar
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15 votes
1 answer
468 views

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 ...
Geoffrey Irving's user avatar
14 votes
2 answers
482 views

What's the bias of random polynomials with low degree over GF(2)?

I have a question concerning low-degree polynomials and probability: What is the (assyptotic behavior of the) probability that a random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
Avishay Tal's user avatar
14 votes
1 answer
338 views

How much independence is required for separate chaining?

If $n$ balls are placed into $n$ bins uniformly at random, the heaviest loaded bin has $O(\lg n/\lg \lg n)$ balls in it with high probability. In "The Power of Simple Tabulation Hashing", Pătraşcu and ...
jbapple's user avatar
  • 11.2k
13 votes
3 answers
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[...
Arthur B's user avatar
  • 429
13 votes
2 answers
377 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, ...
Morgan Shirley's user avatar
13 votes
2 answers
3k views

Sum of Independent Exponential Random Variables

Can we prove a sharp concentration result on the sum of independent exponential random variables, i.e. Let $X_1, \ldots X_r$ be independent random variables such that $Pr(X_i < x) = 1 - e^{-x/\...
NAg's user avatar
  • 666
13 votes
1 answer
286 views

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 $BPP\subseteq \Sigma_2\cap \Pi_2$....
Shaull's user avatar
  • 5,571
13 votes
0 answers
179 views

Generating a random graph with constraints on spectrum

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 ...
Artem Kaznatcheev's user avatar
12 votes
2 answers
2k views

What is the precise definition of Random K-SAT?

There are 4 different constraints we can have when defining Random K-SAT. 1)Total number of literals in a given clauses is exactly K or AT most K 2)A given literal can be used with or without ...
Tayfun Pay's user avatar
  • 2,598
12 votes
1 answer
1k views

Is there a problem in ZPP not yet in P?

Primality was a nice problem that was in ZPP but was not known to be in P. Is there a (preferably simple to state) problem of which we can prove that it is in ZPP but we do not know whether it is in P ...
domotorp's user avatar
  • 13.9k
12 votes
1 answer
521 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 + ...
Clement C.'s user avatar
  • 4,461
12 votes
1 answer
413 views

Measuring the randomness of CNF formulas

It's widely known that CNF formulas can be roughly partitioned in 2 broad classes: random vs. structured. Structured CNF formulas, in opposition to random CNF formulas, exhibit some sort of order, ...
Giorgio Camerani's user avatar
12 votes
0 answers
278 views

Conditional density of primes

We have some theorems about the density of prime numbers, the most famous one is probably the prime number theorem. My question is about the density of primes when we choose random numbers from a ...
Kaveh's user avatar
  • 21.6k
12 votes
0 answers
374 views

Oracle relative to which MA does not have a complete problem?

Babai introduced a hierarchy of complexity classes based on public-coin randomized interactive proof systems, so called Arthur-Merlin games. The game is played by powerful but untrustworthy wizard ...
Dai Le's user avatar
  • 3,664
11 votes
1 answer
726 views

Time complexity analysis for Reingold's UST-CONN algorithm

What is the exact time complexity of the undirected st-connectivity log-space algorithm by Omer Reingold ?
user avatar
11 votes
3 answers
463 views

Randomized algorithms using a stack

I have developed a new derandomization technique which is aimed at recursive randomized algorithms (or) more generally randomized algorithms that use a stack. Unfortunately, I could not find natural ...
Shiva Kintali's user avatar
11 votes
1 answer
422 views

Uniform way of quantifying "branching" in nondeterministic, probabilistic, and quantum computation?

The computation of a nondeterministic Turing machine (NTM) is well known to be representable as a tree of configurations, rooted at the starting configuration. Any transition in the program is ...
Cem Say's user avatar
  • 823
11 votes
1 answer
314 views

Randomized Polynomial Hierarchy?

I wonder, what would happen, if in the definition of $PH$ (Polynomial Hierarchy, see, e.g., here), the role of $NP$ would be replaced by $RP$? It seems, we could still build a hierarchy, the same ...
Andras Farago's user avatar
10 votes
3 answers
646 views

What is the fastest known simulation of BPP using Las Vegas algorithms?

$\mathsf{BPP}$ and $\mathsf{ZPP}$ are two of basic probabilistic complexity classes. $\mathsf{BPP}$ is the class of languages decided by probabilistic polynomial-time Turing algorithms where the ...
echuly's user avatar
  • 549
10 votes
1 answer
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 ...
Thomas Klimpel's user avatar
10 votes
1 answer
267 views

Is deterministic pseudorandomness possibly stronger than randomness in parallel?

Let the class BPNC (the combination of $\mathsf{BPP}$ and $\mathsf{NC}$) be log depth parallel algorithms with bounded error probability and access to a random source (I'm not sure if this has a ...
Geoffrey Irving's user avatar
10 votes
0 answers
200 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 ...
domotorp's user avatar
  • 13.9k
9 votes
1 answer
464 views

Evidence that there is some problem in BQP distinct from BPP?

Are there any evidences (1 physics, 2 mathematics AND 3 computer science) that particular problems such as integer factorization, discrete logarithm are in BQP but not in BPP? There do not seem to be ...
user avatar
9 votes
1 answer
202 views

What is the probability that a random Boolean function has a trivial automorphism group?

Given a Boolean function $f$, we have the automorphism group $Aut(f) = \{\sigma \in S_n\ \mid \forall x, f(\sigma(x)) = f(x) \}$. Are there any known bounds on $Pr_f(Aut(f) \neq 1)$? Is there ...
Samuel Schlesinger's user avatar
9 votes
2 answers
351 views

Almost universal string hashing in $Z_{2^n}$ and sublinear space

Here are two families of hash functions on strings $\vec{x} = \langle x_0 x_1 x_2 \dots x_m \rangle$: For $p$ prime and $x_i \in \mathbb{Z_p}$, $h^1_{a}(\vec{x}) = \sum a^i x_i \bmod p$ for $a \in \...
jbapple's user avatar
  • 11.2k
9 votes
1 answer
322 views

The power of randomized logspace with two-way access to the random tape

Let $\mathsf{ZPL}$/$\mathsf{RL}$/$\mathsf{BPL}$ denote the classes of the languages which are accepted (with zero/one-side/two-side error) by a logspace Turing machine with one-way access to the ...
Hsien-Chih Chang 張顯之's user avatar
9 votes
0 answers
507 views

Statistical relationship between diameter and density in strongly connected random digraphs

I would like to know if there is any known relation between diameter and density in (connected) simple digraphs. Asymptotic results for n→∞, statistical, conditioned results that would restrict the ...
Alex Repain's user avatar
8 votes
8 answers
4k views

Best way to determine if a list of bytes are random?

Is there some algorithm out there that can return some value indicating a level of randomness? I believe it's called Data Entropy. I recently read this article: http://faculty.rhodes.edu/wetzel/...
Corey Ogburn's user avatar
8 votes
2 answers
607 views

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 $\mathbf{P} = \mathbf{BQP}$ ...
Giorgio Camerani's user avatar
8 votes
2 answers
1k views

How to analyze a randomized recursive algorithm?

Consider the following algorithm, where $c$ is a fixed constant. ...
Saeed's user avatar
  • 3,440