# Questions tagged [randomized-algorithms]

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

167 questions
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
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### 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 $j$-...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### How to shuffle colour balls?

I have 400 balls, in which 100 are red, 40 are yellow, 50 are green, 60 are blue, 70 are purple, 80 are black. (balls of the same colour are identical) i need an efficient shuffling algorithm, so ...
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### Is there a randomized algorithm for set-cover?

Is there a well-known randomized algorithm for the set cover problem in the literature - such that it has an approximation ratio of $O(\log n)$ or $f$ - where $f$ is the max frequency of an element. ...
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### 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 ...
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### 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....
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### A converse to Fano's inequality ?

Fano's inequality can be stated in many forms, and one particularly useful one is due (with a minor modification) to Oded Regev: Let $X$ be a random variable, and let $Y = g(X)$ where $g(\cdot)$ is ...
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### 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 ...
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 ...