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Questions tagged [randomized-algorithms]

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

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2 votes
0 answers
87 views

Searching for a proof that non-deterministic logspace with errors is contained in $PL$

In this 10 year old question Non-deterministic logspace with two-sided error the author asked for a complexity class related to $NL$. Namely $NL$ but we are allowed to have two-sided error for the ...
tcs_enjoyer's user avatar
2 votes
1 answer
387 views

A question on combinatorial algorithm

Given n sets $S_1,...,S_n$ such that $S_i \subset \{1,...,n\}$ is there a poly(n) algorithm to find $1 \leq i_1 < i_2 <.... < i_k \leq n$ such that $|\bigcup_{j=1}^k S_{i_j} | = k$ where $1 \...
Rishabh Kothary's user avatar
4 votes
0 answers
90 views

Working out the constants and probabilities of Stockmeyer's approximate counting algorithm

Stockmeyer's 1983 result on approximate counting using a randomness states that if we have some SAT instance $x$ with $C(x)$ satisfying assignments, then we can find the minimum set of $m$ hash ...
Germ's user avatar
  • 191
2 votes
1 answer
147 views

Moments and Deviation (Motwani and Raghavan, 3.7)

Let $a$ and $b$ be chosen independently and uniformly at random from $\mathbb{Z}_n=\{0,1,2,\ldots,n-1\}$, where $n$ is a prime. Suppose we generate $t$ pseudo-random numbers from $\mathbb{Z}_n$ by ...
learning's user avatar
0 votes
0 answers
51 views

Approximation ratio of randomized rounding for integral multi-commodity flow

In [1], Raghavan and Thompson showed that we can use randomized rounding to approximate integral multi-commodity flow and routing with congestion. The result is that suppose the optimal solution is $W$...
Recursion's user avatar
1 vote
0 answers
42 views

How to understand this evolutionary algorithm lower bound calculation?

I have a proof that I understand the most of it except one step Lemma 10. The expected number of steps the $(1+1)$ EA takes to optimize a linear function with all non-zero weights is $\Omega(n \ln n)$....
Edee's user avatar
  • 111
2 votes
0 answers
53 views

Approximately sampling from a discrete unimodal distribution with large support

I have an algorithmic problem and I am curious if a solution is known in the literature, because I cannot find it. I came up with an algorithm of my own, but would be curious if something is known. I ...
user2316602's user avatar
13 votes
2 answers
676 views

What is a very simple pseudodeterministic algorithm (for educational purposes)?

Definition. A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. Question. The notion of a ...
user69687's user avatar
  • 133
0 votes
1 answer
81 views

Generalized assignment problem with overall budget

The problem has N tasks. We have M workers. We have the cost of assigning task i to worker j. We have a profit for assigning task i to worker j. We want to assign each task to exactly one worker. One ...
Exulansis's user avatar
3 votes
0 answers
88 views

Is the Moser-Tardos algorithm used in any real-world applications?

The Moser-Tardos algorithm can be used to construct algorithms for certain combinatorial problems. However, I'm curious about whether this algorithm is utilized in real-world systems (a SAT solver, ...
Lin's user avatar
  • 31
3 votes
0 answers
132 views

Inverse of leftover hash lemma

Leftover hash lemma: Let $X$ be a random variable over $X \in {\mathcal {X}}$ and let $m>0$. Let $h: {\mathcal S} \times {\mathcal X} \rightarrow \{0,1\}^m$ be a 2-universal hash function. If $m \...
delete000's user avatar
  • 828
6 votes
0 answers
135 views

Consistent Sampling a Random Walk

Assume there's a random walk $S_k = X_1 + \dots + X_k$ where $X_i \in \{1, -1\}$ are uniformly iid. I want Alice and Bob to share a function $S(k) = S_k$. A straightforward approach would be to let ...
Thomas Ahle's user avatar
0 votes
1 answer
203 views

Construction of a collection of subsets of $\{1,2,\ldots,n\}$ with certain properties

Let $n$ be a large positive integer. Given a collection $\mathfrak S$ of subsets of $[n] := \{1,2,\ldots,n\}$, and a vector $z=(z_1,\ldots,z_n)\in \{\pm 1\}^n$, define $$ f_{\mathfrak S}(z) := \sum_{\...
dohmatob's user avatar
  • 291
3 votes
2 answers
165 views

Worst-case complexity of computing a certain non-standard dot product + algorithms realizing this complexity

Let $n$ be a large positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an inner-product on $\mathbb R^n$ by \begin{eqnarray} \langle x,y\rangle_{\...
dohmatob's user avatar
  • 291
0 votes
1 answer
146 views

Boosting the probability of success(random projections, johnson lindenstrauss)

In the simple proof of the johnson lindenstrauss lemma written by Sanjoy Dasgupta, Anupam Gupta that can be found here they state the following (p.$62$): Repeating this projection $O(n)$ times can ...
randomizedalgo's user avatar
3 votes
1 answer
225 views

Can the ellipsoid method be used with a randomized separation oracle?

Suppose we are trying to solve the following optimization problem: $$ \text{maximize } ~~ c\cdot y \\ \text{subject to } ~~ y\in S $$ where the region $S$ is described by an exponential number of ...
eden hartman's user avatar
2 votes
0 answers
97 views

Random Self-Reducibility of the Discrete Logarithm

Section 10.1.2 of Sanjeev Arora and Boaz Barak's Computational Complexity: A Modern Approach defines random self-reducibility and proves hardness of the discrete logarithm by reducing a worst case ...
Krish Singal's user avatar
0 votes
0 answers
94 views

Examples of Gaussian randomized algorithms

I've been thinking about algorithms of the form where a quantity $c$ can be viewed as the expectation of some estimator (random variable) $X$ and the expectation is taken over some multivariate ...
user135520's user avatar
1 vote
0 answers
92 views

Generalizing Fano's inequality

Fano's inequality says the following: Theorem: Let $X$ be a random variable with range $M$. Let $\hat{X} = g(Y)$ be the predicted value of $X$ given some transmitted value $Y$, where $g$ is a ...
learning_tcs's user avatar
1 vote
1 answer
67 views

Indexed access with deletion

As part of a larger data structure that I am working on, I have the following sub-problem: I start with $n$ slots in an array. Initially all slots are valid. I want to support two operations: ...
Matthias's user avatar
4 votes
0 answers
84 views

Universal Relation

In the paper Tight Bounds for Lp Samplers, Finding Duplicates in Streams, and Related Problems, the authors consider the universal relation problem in 2-party communication complexity, which is ...
Theo's user avatar
  • 41
2 votes
2 answers
482 views

Trying to understand the intuition behind Yao's Minimax Principle

$\newcommand{\A}{\mathcal{A}}\newcommand{\I}{\mathcal{I}}\newcommand{\E}{\mathbb{E}}\newcommand{\C}[2]{C(I_{#1},A_{#2})}$The question that I am wondering in this post is if there is any intuition to ...
DenLilleMand's user avatar
5 votes
0 answers
137 views

Evaluating arithmetic circuits with stochastic rounding

Let $x_1, \ldots, x_n \in \mathbb{R}$, and let $y = f(x_i)$ be an arithmetic circuit in the $x_i$'s. That is, $f$ is a circuit of negate, add, subtract, and multiply gates. Let $X_i$ be floating ...
Geoffrey Irving's user avatar
11 votes
2 answers
878 views

Randomized algorithms not based on Schwartz-Zippel

Are there any problems that are known to be in a randomized complexity class (e.g. RNC, ZPP, RP, BPP, or even PP), but not in any lower non-randomized class (e.g. NC, P, NP), and whose membership in ...
Shaull's user avatar
  • 5,646
0 votes
1 answer
70 views

Deterministic communication complexity of refinement

A partition of $[n]$ is a collection $\mathcal{P}$ of non-empty subsets of $[n]$ such that for each $i \in [n]$ there is a unique $P \in \mathcal{P}$ with $i \in P$. For partitions $\mathcal{P}, \...
user1868607's user avatar
  • 1,019
5 votes
1 answer
215 views

Hashing-based vs almost uniform sampling-based approximate counting

Corollary 3.6 in the UniqueSAT paper by Valiant and Vazirani [1] states: For any $\varepsilon > 0$ there is a randomized polynomial-time TM with a SAT oracle, which given a SAT formula $f$ outputs ...
delete000's user avatar
  • 828
2 votes
1 answer
105 views

Property testable in sublinear time in bounded degree graphs but not in general graphs

Is there some natural property that is testable in strongly sublinear time (i.e. $O(n^{1-\epsilon})$ for some $\epsilon > 0$) in bounded-degree graphs but not in general graphs? If not such ...
user2316602's user avatar
1 vote
0 answers
52 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. ...
Sudipta Roy's user avatar
4 votes
0 answers
120 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 ...
templatetypedef's user avatar
4 votes
0 answers
155 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 ...
zfkmz's user avatar
  • 187
0 votes
0 answers
62 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 ...
SagarM's user avatar
  • 716
5 votes
2 answers
333 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?
Turbo's user avatar
  • 12.9k
0 votes
0 answers
76 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 ...
Sudipta Roy's user avatar
1 vote
0 answers
93 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 ...
Springberg's user avatar
3 votes
1 answer
227 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 "...
Clement C.'s user avatar
  • 4,471
3 votes
1 answer
446 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 ...
user21820's user avatar
  • 135
0 votes
0 answers
110 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-...
Mahdi Cheraghchi's user avatar
-1 votes
1 answer
108 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\...
Sudipta Roy's user avatar
5 votes
2 answers
250 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 ...
Inuyasha Yagami's user avatar
0 votes
0 answers
47 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 ...
Annonymous's user avatar
1 vote
0 answers
112 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 ...
templatetypedef's user avatar
6 votes
1 answer
263 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 ...
Elle Najt's user avatar
  • 1,469
3 votes
1 answer
202 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 ...
D.W.'s user avatar
  • 12.1k
0 votes
0 answers
112 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 ...
gradstudent's user avatar
  • 1,453
5 votes
0 answers
95 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,...
AmeerJ's user avatar
  • 679
1 vote
0 answers
100 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 ...
gradstudent's user avatar
  • 1,453
1 vote
1 answer
125 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 ...
smapers's user avatar
  • 849
-4 votes
1 answer
510 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 ...
Reihaneh Hassanzadeh's user avatar
2 votes
0 answers
96 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 ...
prsm's user avatar
  • 29
6 votes
1 answer
242 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 ...
Hilder Vitor Lima Pereira's user avatar

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