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Questions tagged [randomized-algorithms]

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

46 questions with no upvoted or accepted answers
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12
votes
0answers
216 views

Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...
11
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0answers
196 views

On preprocessing a convex polyhedron prior to sampling

Most of the algorithms for estimating the volume of a convex polyhedron $K \subset R^d$ assume the existence of an affine transform $T$ with the property that $$ B \subset TK \tilde{\subset}\ \...
9
votes
1answer
309 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 ...
8
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0answers
443 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 ...
7
votes
0answers
540 views

Narrowing the gap between BPP and RP

We do not know yet whether the 2-sided error of $BPP$ allows more computing power than the one sided error of $RP$. In view of derandomization results, the conjectured answer is no, since both classes ...
7
votes
0answers
164 views

Finding the set of paths of smallest cumulated length that cover a given set of patterns

First of all, sorry for this long and maybe not very informative title... Context: Let $G=(V,E)$ be a directed graph, let $v_0 \in V$ be the initial node of paths that I will consider in the graph. ...
6
votes
0answers
968 views

Count $k$-hop neighborhood for every vertex

For a node $v$ of a directed unweighted graph $G$, I define the $k$-hop neighborhood of $v$ as the set of vertices that are reachable from $v$ in $k$ hops or fewer (that is following a path with $k$ ...
6
votes
1answer
549 views

Can this randomized greedy algorithm be made online? Or being proved impossible?

I am going to edge color an undirected simple graph. The following randomized offline algorithm is showed at Online Algorithms for Edge Coloring. Offline: The potential colors are ordered 1, 2, . . ....
5
votes
0answers
146 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 ...
5
votes
0answers
116 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
5
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0answers
224 views

Distributed algorithms on sets

Given a connected arbitrary network $G = (V,E)$, where $V$ is a set of nodes (processors) and $E$ is the set of edges between the nodes. Each node $v _i$ is assigned a non-empty set $S(v _i)$, where $\...
5
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0answers
161 views

Online Interval Coloring Problem

We are given a path $P = (V,E)$ on $n$ nodes, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $...
4
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0answers
144 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.
4
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0answers
105 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 ...
4
votes
0answers
59 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: ...
4
votes
0answers
170 views

Randomized Parallel Algorithm for Maximal Independent Set

There are a couple of randomized parallel algorithms for the maximal independent set problem, e.g. A Simple Parallel Algorithm for the Maximal Independent Set Problem, A fast and simple randomized ...
4
votes
0answers
226 views

Sums of products of bernoulli random variables

Let $x_1 \ldots x_a,y_1 \ldots y_b$ be independent random variables taking values +1 or -1. Consider the sum $$S = \sum_{i,j} x_iy_j.$$ I wish to upper bound the probability $P(|S| > t)$. The ...
4
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0answers
989 views

Lowest Common Ancestor Problem in Directed Acyclic Graphs

What is the current best bound for the following problem in DAG: "For any pair of vertices in a given graph G, return all the LCAs of the same"? Edit: I am working on all-pair reachability problem in ...
4
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0answers
139 views

Concentration Bounds for Dependent Rounding

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
4
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0answers
189 views

Type-and-effect systems, stochasticism and effect squelching: how about quicksort?

There's a feature of Haskell's type system which bugs me: you can't implement a randomized sorting algorithm without the use of randomness spilling out into all of its callers. That seems undesirable....
3
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0answers
156 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 ...
3
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0answers
104 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 ...
3
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0answers
182 views

Probabilistic sorting given pairwise comparison probability

Let $X = \{x_1, \dots, x_n\}$ be a set, and $f:[1..n]^2 \to [0, 1]$ be a function, such that $$f(i, j) \cdot f(j, k) \le f(i, k)$$ For all $1 \le i, j, k \le n$. Does there exist a randomized ...
3
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0answers
284 views

Generating random graphs using the preferential attachment model with degree bounds

I am interested to know the structure of random graphs generated by the preferential attachment model with degree bounds. Specifically, when a vertex is chosen with a probability proportional to its ...
3
votes
0answers
193 views

Randomized rounding on a graph

Assume we are given an arbitrary undirected graph $G = (V, E)$ where $|V| = n$. We are also given real numbers $x_e \in [0, 1]$ for each $e \in E$. These numbers satisfy the following constraint: \...
3
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0answers
282 views

Distinguishing two types of Monte-Carlo algorithms

Recall from Wikipedia that Monte Carlo algorithm is a randomized algorithm whose running time is deterministic, but whose output may be incorrect with a certain (typically small) probability. Consider ...
3
votes
0answers
323 views

Taking Square Roots of Matrices over Z/nZ

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a ...
2
votes
0answers
343 views

the confusion about 'with high probability (w.h.p.)'

w.h.p. can often be seen in the analysis of randomized algorithms. It's definition can be seen here https://en.wikipedia.org/wiki/With_high_probability. However my confusion is that: Assuming we ...
2
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0answers
192 views

Height of randomly built binary search tree by insert and delete?

In Introduction to algorithm (CLRS), even in its third edition (published in 2009) it is noted in Sec 12.4 that little is known about height of randomly built binary search tree using insert and ...
2
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0answers
114 views

What upper bound can we get under 3-wise independence? (comparable edition)

Here is the original question: What bound can we get using $k$-th moment inequality under 3-wise independence? .Yury has given a 3-wise independent example that shows the upper bound is no better than ...
2
votes
0answers
401 views

Texts on application of randomized algorithms

as far as I know, Motwani & Raghavan, ”Randomized Algorithms”, Cambridge University Press, 1995 is the standard book for randomized algorithms. This is an excellent theoretical intro. Which is ...
1
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0answers
69 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 ...
1
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0answers
46 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
0answers
66 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 ...
1
vote
0answers
69 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 ...
1
vote
0answers
55 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....
1
vote
0answers
57 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 ...
1
vote
0answers
81 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, ...
1
vote
0answers
132 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 $k<n$?
1
vote
0answers
76 views

“conservative approximate Set Cover”?

We are given a lattice graph $L$, embedded on the plane, and a certain "shape" (a connected, acyclic subgraph $S$ of $L$). The task is to approximately cover $L$ with translated, rotated and flipped ...
1
vote
0answers
90 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
1
vote
0answers
65 views

Bound the number of rounds in the sampling

Suppose we have a sequence $a_1,a_2,\ldots, a_n$, each $a_i$ is sampled uniformly and independently from $[0,1]$. Define $$ J_1=1,\\ \text{for}~i>1, ~J_i = 1 \iff a_i < \min \{a_1,a_2,...
0
votes
0answers
21 views

Is there a stochastic/online version of the GLM-Tron algorithm?

The GLM-Tron algorithm appeared in Theorem $1$ in this paper, https://arxiv.org/pdf/1104.2018.pdf Is there a stochastic version of this? (...essentially something that will randomly sample a few ...
0
votes
0answers
29 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
221 views

Steiner Tree and minimum spanning tree

If I must connect: $$2^k$$ terminals in a Steiner Tree choosen randomly and connect them with the cheapest component; "loss - contracting algorithm" is a good way? Or is an "Iterative Randomized ...
0
votes
0answers
94 views

To find OR of $\sqrt{n}$ numbers each of $n$ bits?

Given $\sqrt{n}$ numbers of $n$ bits each. I need to find its OR and store it at another number RESULT of $n$ bits. Trivially it can be done in $\mathcal{O}(n \sqrt{n})$ time, or $\mathcal{O}(n \sqrt{...