Questions tagged [probabilistic-algorithms]
Questions about (typically randomized) algorithms that can produce no or an incorrect answer with a certain probability.
203
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Hashing algorithm which minimizes distribution
Consider applications A1, A2 .. AN having properties P11, P12,... PN1, PN2.. Also consider buckets ...
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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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$k$-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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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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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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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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2
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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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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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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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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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1
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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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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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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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1
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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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Understanding simulated annealing information theoretically
So I recently rediscovered simulated annealing through 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 whose ...
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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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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 ...
user97294
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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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1
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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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1
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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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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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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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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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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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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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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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672
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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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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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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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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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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 $x \ne y$ :
$$P_{h \in H_w}(h(x) = h(y)) \...
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What is an example of a weakly universal hash function that is not pairwise independent?
A family of hash functions $H_w$ is said to be weakly universal if for all $x \ne y$ :
$$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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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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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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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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1
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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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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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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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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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1
answer
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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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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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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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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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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 ...
lprndn
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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 match, start over,...
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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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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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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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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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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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