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Questions tagged [probabilistic-algorithms]

Questions about (typically randomized) algorithms that can produce no or an incorrect answer with a certain probability.

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12 views

What is the probability that an expanding bipartite graph exists with the property, |V1|=|V2|?

I want to find a bound on the above problem, and show that a random graph has a positive probability of being an expanding bipartite with the property, |V1|=|V2|. I am not getting, where should I ...
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1answer
44 views

Why an error probability of 1/3 in BPP?

BPP is defined as the class of polynomial-time random algorithms which have an error probability of at most 1/3. But why was 1/3 chosen? If we have an algorithm with some error probability less than ...
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1answer
19 views

Are there efficient probabilistic multiplication algorithms that use O(n log n) gates?

Recently Harvey and Hoeven published a paper proving that integer multiplication can be performed using at most O(n log n) operations. This algorithm is theoretically interesting, but in practice ...
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1answer
22 views

Polysize bounded depth circuit for modified MAJORITY problem

I am trying to show the existence of a polynomial size, bounded depth monotone circuit on the inputs $(x_1,\ldots, x_n)$ that gives $1$ if $\sum x_i \geq n/2 + n/\log n$ and $0$ if $\sum x_i \leq n/2 -...
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1answer
44 views

Can BPP be bounded around any constant other than 1/2?

A language $L$ is in BPP if there exists a randomised TM such that it outputs a correct answer with probability at least $1/2+1/p(n)$ for some polynomial $p(n)$, where $n$ is the length of the input. ...
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20 views

Correctness of Karger's min-cut Algorithm

tl;dr in the analysis for Karger's min-cut, the probability of an edge being in the min-cut in the $j$th iteration, $\frac{k}{0.5k(n-j)}$, neglects the fact that all the edges between the two ...
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30 views

Construction of hash function with a given distribution

Two questions about the construction of a hash function: Let $U = \{u_1,...,u_n\}$ be a set of size $n$, and suppose that one is interested in a function $h\colon U \rightarrow [0,1]$ such that $h$ "...
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22 views

PCP variant in P with non 0 randomness and polynomial proof

I am trying to show that a particular language $L$ in PCP(log,q) is also in P. The PCP protocol works as follows: log many random bits and checks at q positions in a polynomial length proof. The ...
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30 views

Question on preventing k from reducing too quickly during KMV intersection

This question considers KMV, an algorithm that is able to estimate the cardinality (unique item) from a stream of data. The way it does it is to first map the stream of data to a space that almost ...
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1answer
25 views

SVM with a priori information about class probabilities

Given are two 2-d sets, each with its own bivariate normal distribution. I need to build an SVM classifier. The a priori probabilities of each class corresponds to the size of its set (20/50 for the ...
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1answer
66 views

Determine number of values less than mean in one pass through list

The problem statement is as follows: Can we determine precisely the number of elements less than the mean in a list $A$ of $n$ numbers in only one pass through the array (starting at $A_1$ and ...
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1answer
33 views

Singleton in a simple SBM

I can't work out the solution to the following exercise: We have $2n$ vertices grouped in $2$ clusters of equal size. The probability of having an edge between $i$ and $j$ is $p$ if $i$ and $j$ are ...
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1answer
69 views

How does KMV (k minimum value) perform set intersection better than hyperloglog?

In this paper, the author seems to suggest that theta sketches(a variant of kmv) outperforms hyperloglog in cardinality estimation on the intersection of n way streams. ...
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104 views

Perfect Completeness of AM protocols?

I understand the idea behind making a MA protocol perfectly complete. In a MA protocol, Merlin sends a proof $\pi$ which Arthur checks with his machine $V$ by plugging in some random bits $r$ such ...
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1answer
102 views

Probabilistic r-way cut set algorithm

I am reading Probability and Computing, by Mitzenmacher and Upfal, and the exercise 1.24 asks for a generalized algorithm for finding the cut-set of a Graph. In this generalized version, instead of ...
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33 views

Is the definition of $\textbf{BPP}$ robust for doubly exponential small (or even smaller) error?

$\textbf{BPP}$ is usually defined in terms of probabilistic polynomial-time TMs which have an error probability of at most $\frac{1}{3}$. Furthermore, using the Chernoff bound it can be proven that ...
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1answer
63 views

Small space hash functions that are weakly but not strongly universal

This is a follow up to this this question about weakly universal hash functions A family of hash functions $H_w$ is said to be weakly universal if for all $x \ne y$ : $$P_{h \in H_w}(h(x) = h(y)) \...
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1answer
285 views

What is an example of a weakly universal hash function that is not pairwise independent?

A family of hash functions $H_w$ is said to be weakly universal if for all $x \ne y$ : $$P_{h \in H_w}(h(x) = h(y)) \leq 1/m$$ Here the function $h:U \rightarrow [m]$ is chosen uniformly from the ...
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41 views

How to generate a random number in a given range with uniform probability [closed]

I have used programming languages which generate a random number in a given range. Let's say we have a range of 1 to 10 each number has the probability of 1/10 to get selected. What is the criteria ...
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30 views

Probability lower bound on a double cycle on two vertices in random cuckoo graph

I have read Chater 17. Balanced Allocations and Cuckoo Hashing in Mitzenmacher. Upfal. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis and got ...
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49 views

Probability of a double cycle in cuckoo graph

I have read Chater 17. Balanced Allocations and Cuckoo Hashing in Mitzenmacher. Upfal. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis and got ...
2
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1answer
46 views

Why 2 different edge min-cuts in an undirected multigraph must be completely disjoint?

For the proof of a maximum of (n 2) min-cuts in any n-vertex undirected multigraph using the random contraction algorithm, we need to know that no min-cut shares an edge with another different one. ...
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1answer
35 views

HyperLogLog leading zeros distribution

I'm reading this article. In particular I'm having some trouble trying to replicate the results of the first image in that article. For x=9, the graph says that the probability is 0.20 aprox. But ...
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59 views

Distribution of pointer keys in a Skip-list node

Suppose we have a list of $N$ keys where the distribution of keys follows $f(x)$. We construct a skip list over the keys. Now if I pick a key (e.g. 31 in the ...
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1answer
44 views

Concentration bound for sum of dependent geometric random variable?

consider following persudocode: i=0 while(i< k): uniformly pick u,v in V if(uv in E): remove uv form E; i++; let $T$ be the number of ...
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1answer
43 views

Monte Carlo Algorithms : Are there any problems where two opposite Monte Carlo algorithms could solve it?

I started reading on Probabilistic algorithms and Monte-Carlo algorithms. Since a Monte-Carlo can only give a certain answer for either True or False, I was wondering if it's conceivable that for the ...
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2answers
138 views

Randomly Choosing a N-Bit Prime

I've been studying some number theory, and I came across this problem: Lagrange’s prime number theorem states that as N increases, the number of primes less than $N$ is $Θ(N/ log(N))$. ...
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57 views

Insertion in 123 skip list

I had a problem in my exam dealing with 1-2-3 skip lists. I have a problem with the insertion process in a 1-2-3 skip list. What I understand is that if the minimum of heights of 2 nodes in a 123 ...
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21 views

Extracting room voids in a house

I am looking to create a series of closed volumes that represent the empty voids made by rooms in a house. In order to do this, all I have is the raw geometry of all the elements that encapsulate ...
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55 views

Finding subset of integer summing up above threshold

Given an array $|A|=n$ of integers, and $m,k \in \mathbb{N}$, I want to find $m$ elements $a_{i_1},...,a_{i_m}$ of $A$ such that $\sum a_{ij} \geq k$ (repitions allowd), or determine that no such ...
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34 views

Probabilistic linebreaking algorithm

I'm currently trying to implement this paper. Based on a bayesian network, the paper stays unclear about how to ultimately use it's content ("straightforward inference"). But after a lot of ...
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1answer
172 views

Is it possible to simulate a fair coin with a finite number of tossing of a biased one?

It is a classic problem to simulate a fair coin with a biased one. According to Fair Coin (wiki), John von Neumann gave the following procedure: Toss the coin twice. If the results ...
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2answers
78 views

Fast sampling from discrete space

Assume we are given a set $X = \{x_1,...,x_n \}$ of size $n$, and a probability distribution $P$ over $X$. I am interested in an algorithm $A$ which can sample from $X$ according to $P$, i.e. $\Pr(A=...
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1answer
154 views

SAT Solving + Turing Machines

I have a couple of questions based on how SAT solvers work. I understand that SAT solvers may employ any/all of the following techniques: Randomness Heuristics Backtracking SAT is just one example ...
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0answers
45 views

distinguishing two biased coins

I had a simple probability question: suppose we have two coins, coin 1 is heads with probability $= 10\epsilon$ and coin 2 is heads with probability $=\epsilon/10$. Given an unknown coin, how many ...
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1answer
76 views

Given a RxC grid, how to generate N 2D points randomly such that no 3 points are collinear?

Context, I have a geometric algorithm that is sensitive to collinear points and receives as input a list of points in 2D generated randomly. Suppose that I have a Boolean function nonCollinear(x,y,z) ...
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1answer
58 views

Choosing a random edge with restrictions

Given a bipartite graph $G = (V, U, E)$ such that $|V| = |U| =2^n$, one wants to sample an edge from $G$, uniformly at random, with the following operations: 1. One can sample $u \in U$ w.p $\frac{1}{|...
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1answer
53 views

Connection between probabilistic algorithm and distributional algorithm

A basic question about the connection between two notions. I am sure these are known notions in CS but I struggle with the basics: First Definition: A probabilistic algorithm $A(n)$ decides $L$ if for ...
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1answer
77 views

Uniform sampling with constraints

Suppose one wants to uniformly sample a string $w$ of a given length over a finite alphabet, such $w$ satisfies a set of structural constraints (such as - "the third character has to be equal to the ...
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1answer
23 views

Random award question

I'm assisting with the design of an algorithm over the next week to fit the following use case: A person walking in a store has a tablet and approaches possible customers to notify them of a ...
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0answers
13 views

Characterization of P in the context of probabilistically checkable proofs [duplicate]

The PCP theorem characterizes NP as the class of problems checkable probabilistically using log(n) bits and a constant number of random bits and yields $NP=PCP(O(log(n)),O(1))$ . Is there a pair $(f,...
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2answers
1k views

Average Case Analysis for finding max and min value on an array

Given the following algorithm, to find the maximum and minimum values of an array - don't mind the language: ...
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1answer
53 views

Does every language in BPP have a mapping reduction to ATM?

Does every language $C$ in the class $BPP$ have a mapping reduction to $A_{TM}$? $(C\leq _{m} A_{TM})$ $BPP$ is the class of languages that have a probabilistic $TM$ that accepts them with an error $\...
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1answer
41 views

Probability that a leaf with 1 will be selected in game tree evaluation

I am trying to understand randomized AND-OR Game Tree Evaluation. I am stuck with proving the most basic case, namely, an OR node with two leaves (AND node with two leaves is similar). ...
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1answer
50 views

Expected number of nodes in the independent set produced by a coloring algorithm on a graph with maximal degree $k$

We have a graph with $n$ nodes and maximal degree $k$. On this graph we run a coloring algorithm that finds a maximum independent set. The algorithm colors every node green with probability $\frac{1}{...
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62 views

How to select cuckoo filter parameters?

If I want a cuckoo filter to hold a dataset with N entries with a target false positive rate of ϵ, how do I select the sizes for the table, bucket, and fingerprint parameters?
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1answer
100 views

Is there some mathematical properties of the distribution which is the output of a probabilistic polynomial-time Turing machine?

Let $S_{n}$ be an input set whose elements are length $n$, namely $S_{n} = \Sigma^{n}$. For every probabilistic polynomial-time algorithm $A$ and any $u \in \Sigma^{*}$, there exists a function $f^{A}$...
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1answer
58 views

Are probabilisitc algorithms deterministic?

I got confused with deterministic and probabilistic algorithms. Am I right by assuming that algorithms are called probabilistic once they use some sort of randomness? Initially, I thought only ...
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1answer
50 views

If the difference between two oracles is negligible, is the difference between a PPT algorithm with these two oracles also negligible?

We say a negligible function is a function $\epsilon(n):\mathbb{N}\rightarrow \mathbb{R}$ such that for every positive integer $c$ there exists an integer $N_c$ such that for all $n > N_c$, $$\...
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1answer
502 views

Algorithm for finding cliques

Given an arbitrary undirected graph $G = (V,E)$, I am interested in a low-polynomial time algorithm which can find several moderately large (ideally $O(n^\epsilon)$ vertices per clique for $\epsilon &...