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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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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
23 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
61 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
30 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
22 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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85 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
49 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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20 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
48 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
256 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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0answers
33 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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27 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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0answers
43 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
40 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
30 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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0answers
50 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
22 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
27 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
121 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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0answers
52 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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0answers
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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0answers
54 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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30 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
126 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
63 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
83 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
36 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
57 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
51 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
44 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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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
660 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
42 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
39 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
48 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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51 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
98 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
49 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
383 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 &...
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2answers
36 views

Cuckoo filters for non powers-of-2

The Cuckoo filters paper (https://www.cs.cmu.edu/~dga/papers/cuckoo-conext2014.pdf) claims a 95% load factor, however it seems to make an implicit assumption that the table size is a power of 2, and ...
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1answer
89 views

Confuse on using bucket number to replace hashing multiple times in hyperloglog explanation

In this blog the author states that the following So we now have a rather poor estimate of the number of values in the dataset based on bit patterns. How can we improve on it? One ...
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1answer
161 views

Proving $P=BPP$ under a assumption

Let suppose that there exists deterministic polynomial algorithm $A$ that approximates the probability that a given boolean circuit $C$ accepts random input with an error at most $\frac{2}{5}$. More ...
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1answer
120 views

Graph isomorphism in BPP implies it is also in RP

$$L=\{\langle G\rangle \#\langle H\rangle : H, G \text{ are directed isomorphic graphs }\}$$ $\langle G\rangle$ is adjacency matrix written row by row. Show that if $L\in BPP$ then also $L\in RP$. Can ...
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41 views

Show that if language belongs to $BPP$ then also similar language belgons to $QP$

Prove that for each arbitrary set of natural numbers $A$ we have: $$\{bin(n) | n\in A\} \in BPP \to \{0^n|n\in A\}\in QP$$ where $bin (n)$ is binary representation of $n$ $$QP = \bigcup_{c\in \...
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34 views

What's the polynomial involved in the PCP theorem?

Statements of the PCP theorem always speak of a proof of length $poly(n)$. But what polynomial is that exactly? Could you actually construct the PCP for some mathematical fact in real life?
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1answer
184 views

proving $IP^\star = NP$

Consider the following complexity class $IP^\star$, a variant of $IP$. A language $L$ is in $IP^\star$ if there's a proof system $(P,V)$ s.t. $V$ is a verifier runs for a polynomial time and: ...
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53 views

find a values of a array members

We have Some big hashed array of data with boolean values, we know keys, but don't know values. Oracle who have input of array of keys and can tell how much of them are ...