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An algorithm whose behaviour is determined by its input and a generator producing uniformly random numbers.

4
votes
2answers
116 views

Max cut problem between two connected subgraphs

Let $G$ be a connected graph. Consider the problem of finding a partition $G = A \cup B$ into connected subgraphs, so that the cut between $A$ and $B$ is maximized. Is there anything which is known ...
0
votes
0answers
75 views

distNP-complete problem

Here on page 367 there is an example of $\text{dist}\mathbb{NP}$-complete problem: let $U$ contain all tuples $\langle M,x,1^t\rangle$ where there exists a string $y\in \{0,1\}^l$such that the ...
-1
votes
1answer
65 views

Reducing disjoint or indexing or inner-product problem to s-t connectivity problem in directed graph

I am asked to prove that an O(1)-pass randomized streaming algorithm that solves s-t connectivity problem in a simple directed graph $G=(V,E)$ with $|V|=n$ vertices, with sucess possibility $>\frac{...
1
vote
0answers
52 views

Minimization of the maximal adjacent integer sums on a circle

Arrange $\{1,2,\cdots,n\}$ on a circle. What are the arrangements that minimize the maximal sum of all adjacent $k$ integers? For specific and low $n$'s it has been pointed out in math.stackexchange....
4
votes
0answers
131 views

research problem for undergraduate majoring in Computer Science and Mathematics

I am wondering if somebody can suggest a small research problem for undergraduate majoring in Computer Science and Mathematics. Would love to work with random matrices or wavelets.
0
votes
0answers
26 views

Expected size of the min-cut, under edge perturbations

Suppose we have a graph $G(V, E)$. Assume that the min-cut of this graph is given $C=(A/B)$ and denote the size of the cut with $|C|$. Create a random modification of the graph: Drop each ...
0
votes
0answers
57 views

Applications for non-commutative Khinchine inequality

I am looking for the applications of non-commutative Khinchine inequality (see below) in case when Rademacher random variables are tight by the condition $\sum_{i=1}^N\varepsilon_i=M, \, -N \leq M\leq ...
3
votes
1answer
109 views

Tighter Probability Bounds

Let $\mathcal{F}$ be a class of binary functions on a probability space $\Omega$. For $f \in \mathcal{F}$, let $P(f) =\mathbb{E}(f(Z))$ and $P_n(f) = \frac{1}{n} \sum_{i=1}^n f(Z_i)$ where $Z_i$'s are ...
1
vote
1answer
54 views

Does the following 2-rounds distributed algorithm approximates a maximal matching well?

Let $G$ be an undirected graph. I'm looking for a two-rounds distributed algorithm that matches as many vertex pairs as possible. Consider the following protocol for vertex $v$. Use a fair coin to ...
1
vote
0answers
45 views

Generating random labelled trees

I am looking for a simple rejection-free algorithm to uniformly sample random labelled trees (i.e. to generate each of them with the same probability). One possibility is to generate Prüfer sequences ...
5
votes
0answers
103 views

Learning hidden variable distribution

Consider a set of $k$ continuous variables. Each variable $x_k$ is associated with a hidden distribution from which its value is sampled independently of other variables. I am given a set of ...
3
votes
1answer
99 views

Literature reference for search-BPP

I'm trying to find the first article/paper that the complexity class search-BPP appeared in. Search-BPP, as defined as follows (in [1]): A binary relation $R$ is in search-BPP if there is a ...
-3
votes
1answer
85 views

Proving that a random permutation generator is not fair [closed]

If I'm generating random permutations using the following algorithm: ...
9
votes
1answer
308 views

Algorithm to compute distance between powers

Given coprime $a, b$, can you quickly compute $$ \min_{x, y > 0} |a^x - b^y| $$ Here $x, y$ are integers. Obviously taking $x = y = 0$ gives an uninteresting answer; in general how close can these ...
0
votes
2answers
372 views

Naive shuffle algorithm [closed]

Let us have a "shuffle algorithm": for i in range(len(vec)): swap(vec[i], vec[rand()%len(vec)]) Why the reshuffles we get using this algoritm for a number ...
10
votes
1answer
261 views

Find an approximate argmax using only approximate max queries

Consider the following problem. There are $n$ unknown values $v_1, \cdots, v_n \in \mathbb{R}$. The task is to find the index of the largest one using only queries of the following form. A query ...
1
vote
0answers
60 views

Time complexity of polynomial regression with random coefficients

Suppose that I have $$\lim_{k,l,m \rightarrow \infty}(f(x,l) ~~=~~ \sum_{n=1}^{m} g(a_{n,k})h(x,n)$$ where the $a_{n,k}$ are pseudo-random real numbers with $k$ digits generated in an arbitrary way, ...
6
votes
1answer
121 views

Directed graph with bounded in-deg can be partitioned in a balanced way

I want to prove that for all $n$, there exists a constant $c(n)$ such that if $G=(V,E)$ is a directed graph with in-degree bounded by $n$, it is possible to partition the set of vertices $V$ into two ...
8
votes
2answers
253 views

Random point in a d-dimensional ball

I would like to know if there any algorithm to pick a random grid point inside a d-dimensional ball with a given radius R. And if there any algorithm to pick a random arbitrary point inside a d-...
4
votes
1answer
93 views

Sublinear finite-precision sampling in a stream

I am looking to sample a single item from a stream such that each item in the stream has an equal probability of being selected. This is a restricted version of the reservoir sampling problem. On a ...
4
votes
1answer
154 views

Lower bound on probability of getting two close points in a sample of $n$ points

Let $D\subseteq \mathbb{R}^k$. Assume that $Pr_{u,v\leftarrow U_D}[|u-v| <B]>p$ for some $B,p$, where $|u-v|$ is the L1 distance of the vectors. $S\subseteq D$ is obtained by sampling $n$ ...
3
votes
1answer
190 views

Weighted balls and bins

Suppose I have $n$ balls and $n$ bins. Each ball $i$ has weight $w_i$. Let the total weight be $T = \sum_{i=1}^n w_i$. We throw the balls into the bins randomly, i.e., each ball lands into a random ...
10
votes
1answer
171 views

Uniform derandomisation of circuit complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined in the same way as $\textrm{BPP}$ is defined with respect to $\textrm{P}$. ...
12
votes
4answers
472 views

Can we fast generate perfectly uniformly mod 3 or solve NP problem?

To be honest, I don't know that much about how random number are generated (comments are welcome!) but let's assume the following theoretical model: We can get integers uniformly random from $[1,2^n]$ ...
10
votes
1answer
212 views

Randomness and small circuits complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined as $\textrm{BPP}$ with respect to $\textrm{P}$. More formally we provide ...
8
votes
2answers
434 views

Random sampling data structure with removal

I'd like a data structure with the following operations: create a new instances from an array of floating point weights. randomly sample, returning an item with probability proportionate to its ...
13
votes
1answer
286 views

Adleman's theorem over infinite semirings?

Adleman has shown in 1978 that $\mathrm{BPP}\subseteq \mathrm{P/poly}$: if a boolean function $f$ of $n$ variables can be computed by a probabilistic boolean circuit of size $M$, then $f$ can be also ...
11
votes
1answer
381 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 ...
6
votes
1answer
187 views

Finding a positive point for a collection of polynomials

I am wondering about the complexity of the following problem: Given $k$ polynomials $p_1(x_1, \ldots, x_n)$, $p_2(x_1, \ldots, x_n)$, $\ldots$, $p_k(x_1, \ldots, x_n)$ over the $n$ real ...
10
votes
1answer
128 views

Examples of the use of biased estimators

Biased estimators are useful in statistics because they can optimize mean-squared error more than what an unbiased estimator can manage. I was wondering if in theoryCS if there are any very notable ...
1
vote
1answer
76 views

Estimate the maximum frequency of substring with given length in a very long character stream

Suppose there is a very long string $S\in \Sigma^N$ with length $N$, where $\Sigma$ is a relatively small alphabet (for example, $\Sigma=\{'a', 'b', \ldots, 'z'\}$). Now, given a budget $B$, the goal ...
10
votes
1answer
276 views

When does BPP with a biased coin equal standard BPP?

Let a probabilistic Turing machine have access to an unfair coin that comes up heads with probability $p$ (flips are independent). Define $BPP_p$ as the class of languages recognizable by such a ...
3
votes
6answers
364 views

Is there a linear space lower bound for streaming set equality?

Consider two streams. In each stream one string arrives at a time. A query asks: Is the set of strings that has arrived so far the same in both streams? Is there a linear space randomized lower ...
3
votes
1answer
100 views

Quick Sampling from Probability Distribution: Is there a name for this algorithm?

I'm trying to quickly sample from a near-uniform discrete probability distribution exactly once without calculating the entire CDF. Here's the algorithm. Givens: $N,$ the number of elements to ...
2
votes
1answer
347 views

Can every distribution producible by a probabilistic PSpace machine be produced by a PSpace machine with only polynomially many random bits?

Let $M$ be a probabilistic Turing machine with a unary input $n$ whose space is bounded by a polynomial in $n$ and its output is a distribution $D$ over binary strings. Note that the number of ...
1
vote
1answer
98 views

Randomized and deterministic query complexity of symmetric functions

The deterministic query complexity $D(f)$ of a symmetric function $f$ is $\Omega(n)$ (except for f = 0 or f = 1). I am wondering if the same result holds for the (bounded-error) randomized query ...
6
votes
1answer
575 views

Johnson and Lindenstrauss lemma for hamming space

A result of Johnson and Lindenstrauss shows that a set of $n$ points in high dimensional Euclidean space can be mapped into an $O(\frac{\log n}{\epsilon^2})$- dimensional Euclidean space such that the ...
16
votes
3answers
893 views

Examples of successful derandomization from BPP to P

What are some major examples of successful derandomization or at least progress in showing concrete evidence towards $P=BPP$ goal (not the hardness randomness connection)? The only example that comes ...
4
votes
3answers
218 views

Random grid point in a d-dimensional ball

I would like to know if there is any standard algorithm to generate a random grid point inside a d-dimensional ball with a given radius r. Thanks Bin Fu
5
votes
2answers
603 views

The relationship between degree of vertex and size of dominating set

I was wondering is there any relationship between degree of vertex and size of dominating set. For example, if I know the number of vertices is $n$, and I could know each vertex in the graph has ...
7
votes
1answer
214 views

Checking properties of matrices

Given a sparse matrix $A$ in $\mathbb{Z}^{n\times n}$, how easily could one check whether a coefficient $\alpha_k$ of the characteristic polynomial $P_A$ of $A$ is equal to $0$ (without the need to ...
1
vote
0answers
121 views

How to efficiently generate a random 0-1 matrix of a given rank

How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
4
votes
0answers
58 views

Constructing a bad sequence for counter algorithm

Assume that we want to construct a sequence $s\in\{a,b\}^{N}$ such that $s$ contains exactly $n$ times the letter '$a$'. The sequence is then feed to the following probabilistic algorithm: ...
5
votes
1answer
247 views

Smallest $f(n)$ such that $P/f(n) = BPP/f(n)$?

It is well-known that $\mathsf{P/poly}(n) = \mathsf{BPP/poly}(n)$. It is a major open problem to prove the conjecture $\mathsf{P} = \mathsf{BPP}$. $\mathsf{P} = \mathsf{BPP}$ implies $\mathsf{P}/f(n) ...
2
votes
2answers
158 views

Extended version of the paper “Consistent Hashing and Random Trees” with proofs

I've been reading the following paper: David Karger, Eric Lehman, Tom Leighton, Rina Panigrahy, Mathew Levine, Daniel Lewin, "Consistent Hashing and Random Trees: Distributed Caching Protocols for ...
3
votes
0answers
102 views

Number theoretic problems complete for $\mathsf{RL}$

Are there number theoretic problems (such as those related to $\mathsf{gcd}$) that are in $\mathsf{RL}$? Can these also be $\mathsf{RL}$-complete problems (is there any $\mathsf{RL}$-complete ...
8
votes
0answers
389 views

Exponential time hypothesis for random algorithms

The exponential time hypothesis asserts that an algorithm for SAT must take time $2^{\Omega(n)}$. If I am reading this right, this refers only to deterministic algorithms. Is it possible that ETH ...
20
votes
3answers
1k views

Generalizing the “median trick” to higher dimensions?

For randomized algorithms $\mathcal{A}$ taking real values, the "median trick" is a simple way to reduce the probability of failure to any threshold $\delta > 0$, at the cost of only a ...
12
votes
1answer
222 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 ...
-2
votes
1answer
222 views

Is there a randomized complexity class analogous to $\mathsf{P/Poly}$?

$\mathsf{P/Poly}$ captures those problems that could be solved in polynomial time given some precomputed polynomial number of constants. Is there an analogous complexity class in randomized world ...