Questions tagged [probabilistic-algorithms]
Questions about (typically randomized) algorithms that can produce no or an incorrect answer with a certain probability.
138
questions
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1answer
14 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 ...
1
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1answer
20 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 -...
1
vote
1answer
34 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. ...
2
votes
0answers
17 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 ...
0
votes
0answers
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 $h$ "...
0
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0answers
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 ...
0
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0answers
28 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 ...
0
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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 ...
3
votes
1answer
64 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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
49 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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99 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
73 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 ...
3
votes
0answers
32 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 ...
3
votes
1answer
54 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 $x \ne y$ :
$$P_{h \in H_w}(h(x) = h(y)) \...
3
votes
1answer
274 views
What is an example of a weakly universal hash function that is not pairwise independent?
A family of hash functions $H_w$ is said to be weakly universal if for all $x \ne y$ :
$$P_{h \in H_w}(h(x) = h(y)) \leq 1/m$$
Here the function $h:U \rightarrow [m]$ is chosen uniformly from the ...
1
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0answers
40 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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0answers
28 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 ...
2
votes
0answers
46 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
votes
1answer
45 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.
...
2
votes
1answer
34 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 ...
4
votes
0answers
55 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 ...
1
vote
1answer
39 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 ...
2
votes
1answer
41 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 ...
1
vote
2answers
135 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))$.
...
2
votes
0answers
55 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
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 ...
5
votes
1answer
156 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 ...
4
votes
2answers
70 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=...
2
votes
1answer
140 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 ...
1
vote
0answers
41 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 ...
3
votes
1answer
67 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) ...
3
votes
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}{|...
1
vote
1answer
52 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 ...
1
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1answer
62 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 ...
2
votes
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 ...
0
votes
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,...
3
votes
2answers
844 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:
...
1
vote
1answer
52 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 $\...
0
votes
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).
...
0
votes
1answer
49 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}{...
1
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0answers
55 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?
2
votes
1answer
99 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}$...
1
vote
1answer
55 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 ...
1
vote
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$,
$$\...
3
votes
1answer
486 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 &...
2
votes
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 ...
2
votes
1answer
98 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
...
