Questions tagged [randomized-algorithms]
An algorithm whose behaviour is determined by its input and a generator producing uniformly random numbers.
173
questions
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0answers
23 views
Evolving expression trees to approximate an unknown function
I started a small toy project[0], that given a black box function: $\mathbb{R}^n \to \mathbb{R}$ tries to find a good approximation (if not exact solution) by evolving expression trees. These ...
3
votes
1answer
117 views
Binary search on coin heads probability
Let $f:[0,1] \to [0,1]$ be a smooth, monotonically increasing function. I want to find the smallest $x$ such that $f(x) \ge 1/2$.
If I had a way to compute $f(x)$ given $x$, I could simply use ...
0
votes
0answers
40 views
Using martingale arguments to prove convergence of iterative algorithms
Can someone give me typical/educative examples of how martingales can be used to prove convergence of an iterative algorithmS?
The examples I know of can only go so far as to show that there exists ...
5
votes
0answers
58 views
Relaxed minimum dominating set
(I moved this question from cs exchange to here, because it might be more on the topic here)
Let $G=(V,E)$ be a directed graph with $n$ nodes. For a subset of nodes $S\subseteq V$, let $\mathcal{N}(S,...
1
vote
0answers
63 views
What is the state of the art in first order stochastic convex optimization?
What is the optimally fastest convex risk minimizing algorithm which only uses a stochastic first order oracle? Is this SGD?
What is the optimally fastest convex function minimizing algorithm which ...
1
vote
1answer
91 views
Efficient randomness reduction using k-wise independence
Consider a randomized algorithm with runtime $n$, which succeeds with high probability. The algorithm uses $O(n)$ uniformly random bits.
Now it is given that we can replace these uniformly random ...
-4
votes
1answer
83 views
what does NP ⊆ DTIME(…) mean?
Recently I've seen inside theory of a paper. This time complexity, DTIME, is completely new for me. Can somebody explain it?
Also, the paper shows that the misinformation containment problem cannot ...
1
vote
0answers
78 views
Alternative criterion for approximate maximum-weight perfect matching algorithms
I have asked this question in cs.stackeschange, and it was recommended to me that I asked it here.
Is there any literature on approximate maximum-weight perfect matchings where the approximation ...
5
votes
1answer
193 views
Correctness of AKS algorithm for shortest vector problem
Short question
In the end of section 1 of Regev's notes about the AKS algorithm for SVP, why is the following true?
for each such $i$,$y_i− x_i$ remains $w$ with probability $1/2$ or otherwise ...
2
votes
1answer
150 views
How to use a 𝑝-coin so a TM can decide an undecidable language in polynomial time? [closed]
In "Computational complexity- A modern approach" book (page 117) for the lemma 7.12 (following) the author mentioned that if the ρ is efficiently computable ρ-coin cannot give probabilistic algorithm ...
2
votes
1answer
87 views
Robustness to non-uniform randomness vs. one-sidedness
Consider any problem where, fixing two (disjoint) subsets $\mathcal{Y},\mathcal{N}\subseteq \{0,1\}^n$ of the input space, the goal is to obtain a randomized algorithm $D$ which, given a uniformly ...
3
votes
1answer
150 views
Intuitive explanation behind Goemans-Williamson randomized rounding
A very simple randomized cut algorithm achieves $1/2$ of the optimal value: just choose each vertex to be in the cut with probability $1/2$, independently. Goemans-Williamson does something more ...
3
votes
0answers
302 views
Generating a random connected bipartite graph
A (n, m, k)-bipartite graph is a bipartite graphs with:
independent sets of size $\{n, m\}$
a total of $k \geq n+m-1$ edges
We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
11
votes
5answers
1k views
List of quantum-inspired algorithms
Advances in quantum computing have led to the development of new classical algorithms. Notable recent examples are quantum-inspired algorithms for linear algebra:
A quantum-inspired classical ...
1
vote
0answers
48 views
PTAS for projective clustering : survey
$(k,j)$-projective clustering is the natural generalisation for k-clustering, in which one needs to find $k$ $j$-flats in $\mathbb{R}^d$ that minimizes the cost function as defined below:
Given a $j$-...
1
vote
1answer
59 views
What forms of randomness are 'allowed' in FPRASs?
I cannot seem to find a source describing randomized approximation schemes ((F)PRAS) that tells me exactly what sort of randomness a program is allowed to use. A priori it seems to me that being able ...
1
vote
0answers
69 views
Complexity class of approximating perfect match count
We know we can approximate perfect matching count of bipartite and approximate volume of convex bodies in randomized polynomial time.
Is there any evidence these approximations could be in Nick's ...
5
votes
0answers
154 views
Classification of randomized approximation algorithms
Is there a known classification of randomized approximation algorithms, in the same vein as the distinction between Monte Carlo and Las Vegas algorithms for decision problems? (Or equivalently ...
1
vote
0answers
72 views
Missing proof in Salil Vadhan's monograph on pseudorandomness, Random Walks and S-T Connectivity
In Salil Vadhan's monograph on pseudorandomness, chapter 2, half of the proof of Lemma 2.51 is missing
http://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf .
I don't state the full lemma ...
3
votes
1answer
69 views
What is the maximal load of a “latency-bounded” Cuckoo Hash?
Cuckoo Hashing is a method for storing key-value stores (or just a set of keys) with a constant worst-case lookup time.
They use two hash functions $h_1,h_2:\mathbb K\to [n]$, where $\mathbb K$ is ...
5
votes
2answers
262 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 ...
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votes
1answer
117 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
56 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
150 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
31 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 ...
3
votes
1answer
147 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
56 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
59 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 ...
4
votes
0answers
108 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
187 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
140 views
Proving that a random permutation generator is not fair [closed]
If I'm generating random permutations using the following algorithm:
...
9
votes
1answer
312 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
585 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
354 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
98 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
126 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 ...
7
votes
2answers
265 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-...
3
votes
1answer
107 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 ...
3
votes
1answer
162 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 $n$ ...
2
votes
1answer
292 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 ...
9
votes
1answer
210 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 $\textrm{P}$. ...
13
votes
4answers
538 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
238 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 ...
7
votes
2answers
863 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
336 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 ...
10
votes
1answer
409 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 ...
5
votes
1answer
189 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
138 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
113 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
311 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 ...
