Questions tagged [randomized-algorithms]
An algorithm whose behaviour is determined by its input and a generator producing uniformly random numbers.
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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 \...
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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 ...
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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
$$
f_{\mathfrak S}(z) := \sum_{\...
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Worst-case complexity of computing a certain non-standard dot product + algorithms realizing this complexity
Let $n$ be a large positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an inner-product on $\mathbb R^n$ by
\begin{eqnarray}
\langle x,y\rangle_{\...
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Hashing-based vs almost uniform sampling-based approximate counting
Corollary 3.6 in the UniqueSAT paper by Valiant and Vazirani [1] states:
For any $\varepsilon > 0$ there is a randomized polynomial-time TM with a SAT oracle, which given a SAT formula $f$ outputs ...
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Factoring random selfreducibility analogy from discrete logarithm
It is stated in https://en.wikipedia.org/wiki/Random_self-reducibility#Discrete_logarithm that if discrete log is easy for $\frac 1{poly(\log|G|)}$ of all inputs, then discrete log has a randomized ...
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Boosting the probability of success(random projections, johnson lindenstrauss)
In the simple proof of the johnson lindenstrauss lemma written by Sanjoy Dasgupta, Anupam Gupta that can be found here they state the following (p.$62$):
Repeating this projection $O(n)$ times can ...
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Can the ellipsoid method be used with a randomized separation oracle?
Suppose we are trying to solve the following optimization problem:
$$
\text{maximize } ~~ c\cdot y
\\
\text{subject to } ~~ y\in S
$$
where the region $S$ is described by an exponential number of ...
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Using simulated annealing within CSP scheduling problem
I'm looking to use simulated annealing to help optimise the cost of a schedule. The schedule must fit some constraints.
Let's say for the purpose of this question there are two different types of ...
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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 ...
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Deciding DDH based in partial information
Decisional Diffie–Hellman assumption, or DDH in short, is a famous problem in cryptography.
The DDH assumption holds on a cyclic group $(G,*)$ of (prime) order $q$, if for a generator $g \in G$, and ...
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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 ...
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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 ...
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Efficient and simple randomized algorithms where determinism is difficult
I often hear that for many problems we know very elegant randomized algorithms, but no, or only more complicated, deterministic solutions. However, I only know a few examples for this. Most ...
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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:
...
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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 ...
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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 ...
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Generating Graphs with Trivial Automorphisms
I'm revising some cryptographic model. To show its inadequacy, I've devised a contrived protocol based on graph isomorphism.
It is "commonplace" (yet controversial!) to assume the existence ...
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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 ...
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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 ...
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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 ...
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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}, \...
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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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Johnson and Lindenstrauss lemma for hamming space
A result of Johnson and Lindenstrauss shows that a set of $n$ points in high
dimensional Euclidean space can be mapped into an $O(\frac{\log n}{\epsilon^2})$- dimensional Euclidean space such that the ...
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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 ...
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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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Examples of successful derandomization from BPP to P
What are some major examples of successful derandomization or at least progress in showing concrete evidence towards $P=BPP$ goal (not the hardness randomness connection)?
The only example that comes ...
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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 ...
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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 ...
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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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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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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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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 $k$ ...
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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 ...
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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 ...
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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
"...
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In what class are randomized algorithms that err with exactly 25% chance?
Suppose I consider the following variant of BPP, which let us call E(xact)BPP:
A language is in EBPP if there is a polynomial time randomized TG that accepts every word of the language with exactly 3/...
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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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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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Reusing 5-independent hash functions for linear probing
In hash tables that resolve collisions by linear probing, in order to ensure $O(1)$ expected performance, it is both necessary and sufficient that the hash function be from a 5-independent family. (...
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What is the proof of this nonstandard version of Azuma's inequality?
In Appendix B of Boosting and Differential Privacy by Dwork et al., the authors state the following result without proof and refer to it as Azuma's inequality:
Let $C_1, \dots, C_k$ be real-valued ...
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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\...
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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 ...
CommunityBot
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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 ...
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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 ...
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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 ...
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When does randomization speed up algorithms and it "shouldn't"?
Adleman's proof that $BPP$ is contained in $P/poly$ shows that if there is a randomized algorithm for a problem that runs in time $t(n)$ on inputs of size $n$, then there also is a deterministic ...
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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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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 ...
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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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