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

learn more… | top users | synonyms

2
votes
1answer
38 views

Clock solitaire game and principle of deferred decision

I have been reading the randomized algorithm book by Rajeev Motwani and Prabhakar Raghavan. In section 3.5 they have introduced principle of deferred decision which is a different probability space. ...
0
votes
0answers
27 views

Proving that $BPP^{BPP}=BPP$

I'm trying to prove that $BPP^{BPP}=BPP$. $BPP\subseteq BPP^{BPP}$ is obvious. I'm struggling with $BPP^{BPP}\subseteq BPP$.. Can anyone help?
4
votes
2answers
90 views

Are there any useful deterministic quantum algorithms for decision problems?

The vast majority of known interesting quantum algorithms are probabilistic. The only deterministic quantum algorithms that I know of (which aren't trivially equivalent to a classical algorithm) are (...
4
votes
2answers
335 views

Why does Karger's algorithm work “with high probability”

I'm reading Karger 1993's "Global Min-cuts in RNC, and Other Ramifications of a Simple Min-Cut Algorithm" (link). It states that a single round of contractions yields a min-cut with probability $\...
3
votes
1answer
30 views

Replacing n with 2n in asymptotic bounds

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In the proof of the theorem $6$ of the paper on page 632, the authors go on ...
1
vote
1answer
51 views

Why is ZPP = RP ∩ co-RP?

I am trying to prove the theorem that ZPP = RP $\; \cap \; co-RP$. If $L \in \; \subseteq RP \; \cap \; co-RP$ then I can see that it belongs to $ZPP$. But I am unable to prove the reverse direction, ...
3
votes
1answer
52 views

A clarification on $PP$

Wiki in https://en.wikipedia.org/wiki/PP_(complexity) says "a PP algorithm is permitted to do something like the following: On a YES instance, output YES with probability $1/2 + 1/2^n$, where n is ...
1
vote
0answers
50 views

How can we get a Las Vegas algorithm from a Monte Carlo one?

I am trying to solve some exercises on random algorithms from this book, randomized algorithms. This is not a homework. I am only trying to improve my skills. Here is the exercise: Exercise 1.3: ...
3
votes
1answer
68 views

What does the “principle of deferred decisions” formally mean

I have encountered the phrase "Principle of deferred decisions" in Mitzenmacher and Upfal's book on Randomized Algorithms and several other courses online. Isn't it just conditional probability? In my ...
1
vote
1answer
60 views

Probability bounds on size of smaller partition in randomized quicksort

Let $0 < a < 0.5$ be some constant. We have an $n$-element array as input. Randomized quicksort chooses one element from array uniformly at random as a pivot and partitions. With probability $1-...
-1
votes
1answer
40 views

Random quadtrees

I have $N$ uniformly distributed 2D points and I want to find out how many points lie in some small rectangular region. However, the number of points can be arbitrarily large (e.g., $N=10^8$), so I ...
4
votes
1answer
40 views

Logarithmic Randomness is Necessary for PCP Theorem

I am trying to proof the following statement: If $ {\rm SAT} \in {\rm PCP}[r(n),O(1)]$, where $ r(n)=o(\log n)$, then ${\sf P}={\sf NP}$. Here are my ideas for the proof: It can be easily worked ...
3
votes
1answer
656 views

What is the best you can do with a noisy message?

You play a TV game in which you have to open one door out of $n$. Behind each door, there is a treasure with probability 1/2, independent of the other doors, so your apriori chance of winning is 1/2. ...
3
votes
1answer
42 views

Ways to perform “batch” Approximate Member Queries efficiently

In this problem, I'm first given n number of values which I have to store in a space efficient manner. Then I'm given m number ...
1
vote
3answers
117 views

probablistic procedure to permutate array

input : $array[1...n]$ output: permutated array Our algorithm should be probablistic and complexity should be $O(n)$. Could you give me hint ? My weakness is probability theory and it is why I have ...
2
votes
1answer
118 views

Probabilistic algorithm with two-sided error

I am currently studying probabilistic algorithms and came across three major complexity classes: BPP: worst-case polynomial time, two-sided error RP: worst-case polynomial time, one-sided error ZPP: ...
2
votes
0answers
20 views

What is the meaning of the output weights of a Conditional Random Field (CRF) model?

Problem When train my linear chain CRF with annotated observations, I feed it with a number of sequences containing observation values and a "ground-truth" label for each observation. I'm currently ...
0
votes
3answers
59 views

Need for random bits in final PCP theorem statement

PCP theorem states that $$PCP(O(\log n),O(1))=NP.$$ Could we not run through $O(\log n)$ bits deterministically? Does PCP theorem statement mean any set of $O(\log n)$ random bits out of $2^{O(\log ...
3
votes
1answer
91 views

What do we know about $NP \cap co-NP$?

What do we need about the intersection of $NP$ and $co-NP$ apart from the fact that $P$ is a subset of it? (beyond what these answers here say, What do we know about NP ∩ co-NP and its relation to ...
2
votes
1answer
82 views

Can a probabilistic Turing Machine compute an uncomputable number?

Can a probabilistic Turing Machine compute an uncomputable number? My question probably does not make sense, but, that being the case, is there a reasonably simple formal explanation for it. I should ...
1
vote
0answers
26 views

What does ∇log mean in the Robbins-Montro algorithm?

The Robbins/Monro Algorithm is a type of stochastic optimization algorithm of the following form: (as mentioned in wikipedia) $$x_{n+1} - x_n = a_n(\alpha - N(x_n))$$ where $M(x) = \alpha$ is a ...
1
vote
1answer
39 views

Probability of having a log(n) length monotone subsequence in a random permutation of {1,…,n}

How can I compute the probability of having a $\log(n)$ length monotone consecutive subsequence in a random permutation of $\{1,...,n\}$. I wish to upperbound it with $1/n$.
2
votes
1answer
69 views

How do I prove a certain upper bound on the runtime of a probabilistic 2-SAT solver?

As a homework we had to prove a set of upper bounds on a given probabilistic algorithm to find a satisfying assignment for a satisfiable 2-CNF formula. The problem is reproduced below. I'm sorry for ...
-1
votes
1answer
134 views

Understanding polynomial equality testing using randomized algorithms

A file is downloaded from a server and is represented as $a = \{0, 1\}^n$. The server has that file as $b = \{0, 1\}^n$. We want to ensure a degree of certainty that $a=b$, using a randomized ...
1
vote
1answer
106 views

Anagrams solver based on transitions probability

I have an English dictionary (text file) and the frequency of 2-grams, 3-grams and 4-grams as the beginning of each word. I need to write an algorithm that, with a given word, calculates the possible ...
1
vote
1answer
393 views

3/2 - Approximation probabilistic algorithm for MAX-3-COLOR

I have a textbook question here regarding Max-3-Coloring and need some assistance with it. I have searched for any type of information regarding it but haven't found anything substantial. Here is the ...
0
votes
1answer
39 views

How does these Probing time occurs for hash tables

I am having a hard time understanding the numbers of probing which might occur due to using different collision prevention method such as separate chaining, Linear Probing, double probing, which is ...
4
votes
1answer
467 views

What is With High Probability on Probabilistic Algorithms?

I watched lecture from MIT about Skip List. Overall, I understand the material, but one thing. What is "with-high-probability"? I really don't get it at all. I've seen the lecture notes but still didn'...
1
vote
0answers
50 views

Using “incremental algorithms” to find the $k^{th}$ smallest number

This is what I vaguely understand of what an "incremental algorithm" is - say one such for calculating the $k^{th}$ smallest number for a given sequence of elements $x_1, x_2,...,x_n$ then after the ...
3
votes
2answers
64 views

About sorting numbers in linear time

If one is given $n$ numbers picked uniformly at random from the interval $[0,1]$ then is it possible to sort them in linear time? It seems to me that some such method exists which uses binary ...
4
votes
1answer
131 views

Understanding the Sipser-Gacs-Lautemann theorem

The class $BPP$ contains all the languages decided by a probabilistic Turing machine in polynomial time with probability of success more that 2/3 for every input. The class $\Sigma^p_2$ contains all ...
5
votes
2answers
160 views

Choosing error rates for probabilistic algorithms

Probabilistic algorithms often have a parameter that allows one to tune the error rate, typically by running the algorithm repeatedly. This often gives an error rate of something like $2^{-k}$ for $k$ ...
11
votes
1answer
159 views

Can an NP-hard problem be polynomial on average?

I'm wondering if there are any $NP$-hard problems which are ``polynomial" in the average case. I think there are two ways to interpret this? If $P \neq NP$, can there be an algorithm solving an $NP$-...
10
votes
2answers
262 views

Would $\sf RP = NP$ imply $\sf NP = coNP$?

If $\sf RP = NP$ then the hierarchy collapses to its second level (by the Karp-Lipton theorem). But what about $\sf NP$ and $\sf coNP$? I tried to prove that $\sf BPP$ is contained in $\sf NP$ (the ...
9
votes
2answers
210 views

Probability Distributions and Computational Complexity

This question is about the intersection of probability theory and computational complexity. One key observation is that some distributions are easier to generate than others. For example, the problem ...
2
votes
1answer
99 views

Bloom filter variant

I've been playing around with a simple probabilistic data structure which is very similar to a Bloom filter. Where a Bloom filter would use $k$ independent hash functions to choose $k$ of the $m$ bits ...
2
votes
1answer
394 views

Expected maximum bin load, for balls in bins with equal number of balls and bins [closed]

Suppose we have $n$ balls and $n$ bins. We put the balls into the bins randomly. If we count the maximum number of balls in any bin, the expected value of this is $\Theta(\ln n/\ln\ln n)$. How can we ...
2
votes
1answer
89 views

Are there any practical differences between a Turing machine with a PRNG and a probabilistic Turing machine?

Say I plugged in a hardware true-random number generator (TRNG) to my computer, then wrote programs with output that depends on the TRNG's output. Can it do anything non-trivial that a Turing machine ...
10
votes
0answers
211 views

Proof of PCP theorem

I am reading the proof of PCP theorem in Proof Verication and Hardness of Approximation Problems. The following paragraph appears in section 3 (page 4), "Outline of the Proof of the Main Theorem". ...
1
vote
0answers
160 views

PageRank and EigenTrust: How small should epsilon be?

For probabilistic algorithms such as PageRank and EigenTrust, the stopping case is given as $|R_{t+1} - R_{t}| < \epsilon$ (i.e. convergence is assumed). Neither the papers on EigenTrust or ...
1
vote
1answer
199 views

How are Bayesian Nets, Hidden Markov Chains, Conditional Random Fields and Neural Nets related?

I am having an AI exam in two weeks, and I am still figuring out certain concepts and ideas, related to Bayesian Nets, Hidden Markov Chains, Conditional Random Fields and Neural Nets (yes it is all ...
5
votes
1answer
140 views

Possible to construct a probabilistic halting problem solver?

I'm a CS undergrad so my math/CS knowledge is not that deep so please correct me if my premise is flawed or I have made some incorrect assumptions. So I was thinking, much in the way that some ...
8
votes
0answers
106 views

Complexity class for probabilistic approximation algorithms with bounded error

What's the name of a complexity class of optimization problems that have "bounded error probabilistic approximation algorithms"? Bounded error probabilistic version of APX (as BPP is bounded error ...
5
votes
1answer
161 views

Deleting in Bloom Filters

I know that standard Bloom Filters only have operations like inserting elements and checking if an element belongs to filter, but are also some modification of Bloom filters which enable a delete ...
2
votes
2answers
207 views

What would be a decent threshold for classification problem?

I'm using machine-learning algorithms to solve binary classification problem (i.e. classification can be 'good' or 'bad'). I'm using SVM based algorithms, ...
2
votes
1answer
44 views

finding a an object with constant velocity on an infinite grid in discrete time steps

Assume you have an object moving at a constant velocity(up, down, left, right) in a grid. You have unlimited resources (memory, time). At any given time step in the grid, you can "guess" the location ...
4
votes
2answers
969 views

why not just use a random number generator as a hash function?

I guess at the heart of this is that I don't really understand hash functions. One article says any function mapping objects to an object of fixed size: A hash function usually means a function ...
7
votes
3answers
289 views

Computer science problems related to music?

Are there any CS problems, preferably open, that are related to music or musical theory somehow? I would think of problem with musical notation but also probabilities when randomizing according to a ...
4
votes
3answers
322 views

Why do Bloom filters work?

Let's say I am using Bloom filters to create a function to check if a word exists in a document or not. If I pick a hash function to fill out a bit bucket for all words in my document. Then if for a ...
6
votes
2answers
586 views

Bloom filter and perfect hashing

A Bloom filter uses a hash function to test membership in a given set $S$, by checking if an item is present of not at the specified position. To mitigate the effect of hash collision, multiple ...