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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21 views

Probability of loss using a binary symmetric channel

Today we talked about Information Theory and the binary symetric channel. For newbies here is a little explanation : For instance if I want to send a binary to someone : The bit will be "flipped&...
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11 views

Populating a vector of numbers to expose an error in a function implementation

So lets say I'm writing an algorithm that takes a vector as input. I want to know that I'm writing this algorithm correctly however so I of course write tests to see if the output equals what I expect ...
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22 views

Distributional error probability of deterministic algorithm implies error probability of randomized algorithm?

Consider some problem $P$ and let's assume we sample the problem instance u.a.r. from some set $I$. Let $p$ be a lower bound on the distributional error of a deterministic algorithm on $I$, i.e., ...
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13 views

Hash function, $h(k) = \lfloor km \rfloor$ is simple uniform for real $k$ independently, uniformly distributed in the range $0 \leq k < 1$

I was going through the text Introduction to Algorithms by Cormen et. al. where I came across the following statement: If the keys are known to be random real numbers $k$ independently and uniformly ...
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1answer
30 views

Confusion about the Hiring Problem

I'm confused about where the probability from the hiring problem comes from. For background: We interview one person everyday who has a quality characteristic, x, from 0 to 1(distributed uniformly). ...
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30 views

Why Convolutional codes is easy to factor/handle the uncertainty of a bit being a 0 or a 1 into the decoding?

Here is an excerpt from Andrew S. Tanenbaum, Computer Networks, 5th edition, Chapter 3 (The data link layer), Page 208: My question is about this part. [Convolutional codes have been popular in ...
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1answer
112 views

What is an example of a Monte-Carlo algorithm for finding a Hamiltonian path?

I've recently been made aware that there exist Monte-Carlo algorithm(s?) for determining whether a Hamiltonian path exists in a graph. I am struggling to figure out how it would work. What is the ...
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17 views

Reference on PCP theorem

Can someone recommend a reference on the PCP theorem that explains how it works via examples rather than formal CS language? Specifically, I'm looking for the example on how graph colouring can be ...
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45 views

Hashing algorithm which minimizes distribution

Consider applications A1, A2 .. AN having properties P11, P12,... PN1, PN2.. Also consider buckets ...
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48 views

Sizing a Cuckoo filter for a subset of elements

Question I am considering the use of Cuckoo filter for a business case. To simplify the explanation here is an analogy of my needs: There are over $n = 30 000$ first names that exists in the whole ...
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1answer
95 views

$k$-coloring in BPP, implies $k$-coloring in ZPP

Consider the next problem: $k$-COL: Given a graph $G=(V,E)$, does it have a valid $k$-coloring? I need to prove that if $k$-COL is in BPP, then it is also in ZPP. In other words, show that if there ...
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45 views

Is there a complexity class QPP?

The complexity class PP is not considered tractable, because the probability of success can get arbitrarily close to 50% from above as the problem instances get larger, so that (e.g. if this approach ...
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27 views

Recover a matrix with minimum number of queries

Alice has a matrix $A \in \{0,1\}^{n \times m}$ such that the sum of each row is $1$. Bob tries to find the indices of the ones (he knows that the sum of each row is $1$). The type of questions Bob ...
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1answer
46 views

Probability of terminating in a state in a probabilistic algorithm

Suppose i have a circular array of $n$ elements. At time $t=0$ i am in position 0 of the array. The algorithm moves left or right with probability $p=1/2$ (since the array is circular when it moves ...
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1answer
31 views

Demonstrating that probability for every possible result is uniform at the end of an algorithm

I have memory of $k$ elements that you can imagine being represented by an array. One by one, the array receives a value corresponding to the time index, for example at $t=1$ the value will be $1$. At ...
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23 views

Chernoff Bounds (upper tail)

For the proof of Chernoff Bounds (upper tail) we suppose δ<2e−1 . Like in this paper ([see this link ]) 1. Can you tell me why ?
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71 views

Prove an estimator

Consider an undirected graph $G=(V,E)$ representing the social network of friendship/trust between students. We would like to form teams of three students that know each other. The question is to ...
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20 views

Find expectation and calculate Chernoff bound [duplicate]

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same 2 tasks to every ...
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1answer
286 views

Find expectation with Chernoff bound

We have a group of employees and their company will assign a prize to as many employees as possible by finding the ones probably better than the rest. The company assigned the same $2$ tasks to every ...
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1answer
575 views

Derandomization of vertex cover algorithm

I have the following randomized-algorithm for the vertex cover problem. Let $B_0$ be the output set: Fix some order $e_1, e_2,...,e_m$ over all edges in the edge set E of G, and set $B_0=∅$. Add to ...
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104 views

Grover's algorithm on probabilistic classical machines

As a starting point for this question, I came across this question, which AIUI is citing a construction showing how to simulate quantum circuits with a $PP$ algorithm, i.e. implying quantum ...
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1answer
16 views

Predicting the outcome of sporting events with multiplicative scoring

In the Olympic format for sport climbing, eight athletes compete in three rounds of climbing. Their final score is the multiplication of their rankings in each round. For example, an athlete who comes ...
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1answer
58 views

Understanding simulated annealing information theoretically

So I recently rediscovered simulated annealing though a path that others seem to be well aware of. I was aware of Metropolis-Hastings as a sampling algorithm that creates a Markov-Chain who's ...
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14 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
105 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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45 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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35 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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98 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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41 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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34 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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28 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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1answer
32 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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67 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
36 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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302 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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190 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
241 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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38 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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112 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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447 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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44 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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32 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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59 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 ...
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1answer
89 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
40 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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75 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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74 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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62 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
209 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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22 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 ...