Questions tagged [randomized-algorithms]
An algorithm whose behaviour is determined by its input and a generator producing uniformly random numbers.
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47 views
Concentration bounds for hypergeometric distribution
I asked a similar question a while back. I have reformulated the question. My original intention was to ask this question.
Suppose we have an urn containing $N$ balls, $M$ of which are red, rest are ...
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0answers
61 views
A variant of “hypergeometric” distribution
This is already posted here.
We all are aware of Hypergeometric distribution. Let me first briefly discuss what it is.
Suppose we have an urn containing $N$ balls, $M$ of which are red, rest are blue. ...
1
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0answers
48 views
Failing to understand a lemma regarding Robust Low Rank Approximation
I am reading Low Rank Approximation in the Presence of
Outliers by Bhaskara and Kumar and kind of stuck at the proof of Lemma 9. The paper studies robust (to outliers) low rank approximation problem. ...
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0answers
130 views
Removing k vertices to obtain a Cycle Graph
I am not sure if this is a known problem or not.
For an undirected graph, $G$ and $k\in N$ we need to decide if there is a subset $S\subseteq V(G)$ of size at most k s.t every connected component in $...
4
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0answers
81 views
Simple randomized priority queue matching the Fibonacci heap time bounds?
Since the Fibonacci heap was developed, many other priority queues have been invented with equivalent time bounds and a simpler design (e.g. hollow heaps, quake heaps, etc.).
Many classical worst-case ...
0
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0answers
80 views
Extension of a sampling Lemma of Indyk for $k$-median cost
A famous sampling lemma of Indyk (Theorem 31 here) states that if the $k$-median cost of a finite point set(in a metric space) is large with respect to the optimum, then this is true for the case of a ...
4
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0answers
99 views
Are there any known Chernoff/Hoeffding bounds for the case of “almost independence”?
The usual statement of a Hoeffding bound (e.g. https://sites.math.washington.edu/~morrow/335_17/ineq.pdf) requires independent random variables.
My question is: Do there exist bounds similar to ...
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0answers
53 views
Reference Request : Accessible reference for Randomised algorithms and Hashing for non-Computer Scientists?
My goal is to understand well a paper like ApproxMC. It discusses the use of Hash functions for Propositional Model Counting. In my understanding what they call hash functions are just random XOR's ...
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0answers
25 views
Given an input sequence of real numbers, how to find the closest sequence in a large set of sequences
We are given a set $S$ of $m\gg 1$ sequences (arrays) of $n$ elements, where each sequence $s\in S$ belongs to $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a ...
4
votes
2answers
199 views
Is there a linear time algorithm for integer multiplication verification?
There is a quadratic randomized algorithm for matrix product verification.
Is there a similar trick to 'verify given three integers $n,a,b$ if $n=ab$ holds?' in $O(\log n)$ time?
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47 views
A variant of randomized co-ordinate descent
Let us consider the following optimization problem.
$\mathcal{P} =\{P_1,\cdots,P_n\}$, where $P_i\subset\mathbb{R}^d$. Let $m = max_i\lvert P_i\rvert$. The goal is to find a point $c$ such that ...
0
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0answers
80 views
Optimal $\ell^1$ sketching, including the coefficient
Let us define the sketch as mapping $\varphi:R^D\to R^d$, such that for arbitrary $x\in\mathbb{R}^D$, its $\ell^1$ norm is preserved up to $\epsilon$ error, with $1-\delta$ success probability:
$$\...
1
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0answers
69 views
Cover Time of Random Walks with Backtracking on Directed Graphs
The cover time of a random walk on an undirected graph is the expected time for the walk to visit all vertices of the graph (starting from an arbitrary vertex). It is well known that any connected ...
3
votes
1answer
122 views
Converting a Bernoulli to a Gaussian
It is not hard to see that, given one sample from a univariate unit-variance Gaussian $X\sim \mathcal{N}(\mu,1)$ with unknown $\mu$ s.t. $0<|\mu|\leq 1$, one can simulate one draw from a
"...
3
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1answer
312 views
Can the halting problem be solved probabilistically? [closed]
Let $H$ be the halting oracle, meaning that $H$ is a function on pairs of strings such that $H(P,X) = 1$ iff $P$ halts on $X$. A probabilistic program is a program that has (oracle) access to a random ...
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0answers
84 views
Efficient sampling of primes
Using rejection sampling, it is trivial to construct a Las Vegas algorithm for sampling a uniformly random prime number less than a given $N$. What is known about sampling algorithms that run in worst-...
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1answer
102 views
Required sample size to hit certain subset of a ground set
Suppose $X$ is a set of $n$ points in $\mathbb{R}^d$ and $N_1,\cdots,N_k$ are k disjoint (unknown)subsets of $X$. There is a probability distribution $\phi$ on $X$ defined as $\phi(p) = \frac{\lvert\...
4
votes
2answers
164 views
kmeans++ for arbitrary metric spaces and general potential function
I was reading this popular paper "k-means++: The Advantages of Careful Seeding". It appeared in SODA 2007. Since this technique is the most popular clustering technique, I am hoping that my question ...
0
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0answers
43 views
understanding generalized coupon collector for distributions or learning mixture of distribution
Lets suppose we have a set $S=\{1,\ldots,n\}$ and $P$ is the uniform distribution over two subsets $T_1,T_2\subseteq S$, each of size $m\leq n/100$. Now, suppose somehow is given uniform samples from ...
1
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0answers
65 views
Sampling from a family of hash functions, not uniformly at random?
Many algorithms and data structures assume access to a family of hash functions satisfying some nice property (say, $k$-independence or $k$-universality). In these cases, we usually assume that we ...
6
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1answer
174 views
Uniformly sampling or counting connected graph partitions with any number of blocks
Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard?
To clarify: By a ...
3
votes
1answer
142 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
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0answers
55 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
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0answers
74 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,...
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0answers
70 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
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1answer
101 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 ...
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votes
1answer
205 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 ...
2
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0answers
86 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
201 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
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1answer
230 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
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1answer
95 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 ...
4
votes
1answer
227 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 ...
4
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0answers
579 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 ...
12
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5answers
2k 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 ...
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0answers
52 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 $j$-...
0
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1answer
61 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 ...
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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
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0answers
161 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
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0answers
83 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
79 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
386 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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1answer
162 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
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0answers
57 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
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0answers
158 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
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0answers
39 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
170 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
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1answer
57 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
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0answers
75 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
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0answers
111 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
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1answer
231 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 ...
