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

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0answers
24 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 ...
2
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2answers
40 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
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1answer
44 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
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2answers
79 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 $k$ ...
9
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1answer
114 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 ...
8
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0answers
80 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 ...
8
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1answer
95 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
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1answer
49 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
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1answer
123 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
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1answer
47 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 ...
7
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0answers
111 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". ...
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0answers
57 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
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1answer
100 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
84 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 ...
6
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0answers
88 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
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1answer
109 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 ...
0
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1answer
80 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
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1answer
32 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 ...
3
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2answers
249 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 ...
6
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3answers
195 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
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3answers
269 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 ...
5
votes
1answer
237 views

Bloom filter and perfect hashing

Bloom filter use a hash function to test membership for S by checking if an item is present of not at the specified position. To mitigate the effect of hash collision, multiple functions are used, ...
4
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0answers
48 views

Signal-based Search

This is more of an open-ended information question, but to make it concrete, here's an example problem I have thought up: Consider an $N\times N$ grid, $N$ odd, and consider that a single chunk of ...
1
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1answer
71 views

How to sample uniformly from a stream of elements, some of which are unsuited?

I get values $x_t$ in an online fashion and want to buy "good" ones, where "good" means that some measure $P(x_t) >T$. Consider the following simple algorithm. ...
1
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1answer
508 views

How to get expected running time of hash table?

If I have a hash table of 1000 slots, and I have an array of n numbers. I want to check if there are any repeats in the array of n numbers. The best way to do this that I can think of is storing it in ...
7
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2answers
156 views

Are probabilistic search data structures useful?

A SkipList provides the same $O(\log n)$ bounds for search as a balanced tree with the advantage that rebalancing isn't necessary. Since the SkipList is constructed using random coin flips, these ...
2
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0answers
43 views

Approximate target subset by intersecting other subsets

Let $S$ be a finite set of integers (this set contains about 200000 elements). Let $T \subset S$ be a particular subset of $S$ called target. $S$ keeps growing. So does $T$. Each new element of $S$ ...
1
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1answer
53 views

BPP clarification

I know that $\sf BPP[2/3,1/3]= BPP[\alpha,\beta]$ when $\alpha\lt\beta$, but I read something on Wikipedia which got me confused: In practice, an error probability of $1/3$ might not be ...
8
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1answer
310 views

Prove or refute: BPP(0.90,0.95) = BPP

I'd really like your help with the proving or refuting the following claim: $BPP(0.90,0.95)=BPP$. In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time ...
6
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1answer
127 views

BPP search: what does boosting correctness entail?

It is not really clear to me, how and if I can do boosting for correctness (or error reduction) on a BPP (bounded-error probabilistic polynomial-time) search problem. Can anyone of you explain me how ...
3
votes
2answers
453 views

Bloom Filter for 208 million URLs

I need to create a bloom filter of 208 million URLs. What would be a good choice of bit vector size and number of hash functions? I tried a bit vector of size 1 GB and 4 hash functions, but it ...
4
votes
1answer
105 views

What is known about coRL and RL?

Wondering about any known relations between $\mathsf{RL}$ complexity class (one sided error with logarithmic space) and its complementary class, $\mathsf{coRL}$. Are they the same class? What are ...
5
votes
2answers
118 views

An edge that connects more than two nodes in a graph?

Is there a way to create a single edge on a graph that connects 3 or more nodes? For example, let's say that the probability of Y occurring after X is 0.1, and the probability of Z occurring after Y ...
6
votes
1answer
125 views

Probabilistic poly-time machine always halts on all inputs?

In the usual definition of probabilistic poly-time machine it is said that the machine halts in polynomial time for all inputs. Is the intention really to say that the machine halts for all inputs, ...
7
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1answer
271 views

Probabilistic test of matrix multiplication with one-sided error

Given three matrices $A, B,C \in \mathbb{Z}^{n \times n}$ we want to test whether $AB \neq C$. Assume that the arithmetic operations $+$ and $-$ take constant time when applied to numbers from ...
3
votes
1answer
116 views

Randomized String Searching

I need to detect whether a binary pattern $P$ of length $m$ occurs in a binary text $T$ of length $n$ where $m < n$. I want to state an algorithm that runs in time $O(n)$ where we assume that ...
4
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2answers
139 views

Is the number of coin tosses of a probabilistic Turing machine a Blum complexity measure?

I read that the number of coin tosses of a probabilistic Turing machine (PTM) is not a Blum complexity measure. Why? Clarification: Note that since the execution of the machine is not deterministic, ...