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

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31 views

What is the quantity sold for a specific fruit & country combination?

What is the algorithm that generates these potential quantities that meet the given criteria? Essentially - there are number of quantities for a fruit and country combination. E.g: ...
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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 ...
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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 ...
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0answers
30 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 ...
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1answer
92 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 ...
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2answers
54 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=...
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1answer
35 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 ...
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0answers
30 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 ...
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1answer
44 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
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1answer
55 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}{|...
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1answer
42 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 ...
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1answer
35 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 ...
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1answer
22 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 ...
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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,...
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2answers
239 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: ...
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1answer
30 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 $\...
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0answers
19 views

Distributed systems, finding N-th lightest weight in graph

I have a complete graph with N vertices, with positive weights assigned to their edges. Each node knows the weight of its edges. Each node has a unique ID. We can send message on each edge, one ...
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1answer
37 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). ...
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1answer
27 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}{...
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0answers
34 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?
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0answers
35 views

Can I union two cuckoo filters?

If I have a cuckoo filter containing {a,b} and another cuckoo filter containing {a,c}, can I union the filters to make one ...
2
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1answer
92 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 $f^{A}$...
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1answer
34 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 ...
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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 $n > N_c$, $$\...
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1answer
127 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 &...
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10 views

Can the following be viewed as a separation of complexity classes?

Suppose I'm trying to simulate a coin flip with probability $p$ of landing heads. But I'm only given access to perfectly random bits where each outcome is equally likely. If $p$ is a nice number like $...
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2answers
35 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
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1answer
78 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 ...
2
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1answer
156 views

Proving $P=BPP$ under a assumption

Let suppose that there exists deterministic polynomial algorithm $A$ that approximates the probability that a given boolean circuit $C$ accepts random input with an error at most $\frac{2}{5}$. More ...
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1answer
110 views

Graph isomorphism in BPP implies it is also in RP

$$L=\{\langle G\rangle \#\langle H\rangle : H, G \text{ are directed isomorphic graphs }\}$$ $\langle G\rangle$ is adjacency matrix written row by row. Show that if $L\in BPP$ then also $L\in RP$. Can ...
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0answers
37 views

Show that if language belongs to $BPP$ then also similar language belgons to $QP$

Prove that for each arbitrary set of natural numbers $A$ we have: $$\{bin(n) | n\in A\} \in BPP \to \{0^n|n\in A\}\in QP$$ where $bin (n)$ is binary representation of $n$ $$QP = \bigcup_{c\in \...
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32 views

What's the polynomial involved in the PCP theorem?

Statements of the PCP theorem always speak of a proof of length $poly(n)$. But what polynomial is that exactly? Could you actually construct the PCP for some mathematical fact in real life?
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1answer
100 views

proving $IP^\star = NP$

Consider the following complexity class $IP^\star$, a variant of $IP$. A language $L$ is in $IP^\star$ if there's a proof system $(P,V)$ s.t. $V$ is a verifier runs for a polynomial time and: ...
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0answers
53 views

find a values of a array members

We have Some big hashed array of data with boolean values, we know keys, but don't know values. Oracle who have input of array of keys and can tell how much of them are ...
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1answer
103 views

What does it mean for a problem to be solved in polynomial time “relative to” an oracle?

I came across the following theorem in page 12 of the following pdf : There exists an oracle relative to which there is a problem solvable in polynomial time (with bounded error probability) on a ...
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1answer
33 views

Examples of difference between Hidden Markov Model and Bayesian Network?

I am trying to more deeply understand the difference between Hidden Markov Models and Bayesian Network? The general idea is that HMMs have a single variable which has probabilities of entering ...
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1answer
52 views

Is $A_{PTM}$ a BPP-complete language?

The language $A_{PTM}$ is defined as the acceptance problem on a Probabilistic Turing machine. $A_{PTM}=$ { $<M, x> | M$ on input $x$ accepts with an error probability less than or equal to 1/...
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0answers
53 views

Stochastic Computing: What is “Bundle Processing”?

I'm puzzled by a short paragraph found in the article on Stochastic Processing in Wikipedia. There it says: Bundle Processing involves sending a fixed number of bits instead of a stream. One of the ...
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1answer
70 views

Does a computation tree for a probabilistic Turing machine have to be a binary tree?

Ever example of computation on a probabilistic Turing machine that I have read uses only a binary tree to show the computational branches of a probabilistic Turing machine. I know that an NTM may use ...
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1answer
30 views

Using amplification lemma to decrease error probability gives 0 as divisor

I am attempting to work through an example of how to select an error bound, and then determine the number of simulations necessary by the amplification lemma to obtain the desired error bound. I have ...
4
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1answer
40 views

Hidden Markov Model initial probability reestimate: Why $\pi^*_i = \gamma_i(1)$ instead of $\pi^*_i = \frac{\gamma_i(1)}{\sum_{j = 1}^N \gamma_j(1)}$

In the sources I consulted it states that in the Baum Welch algorithm the reestimate of the initial probability of state $i$ of the HMM is $\pi^*_i = \gamma_i(1)$. But $\gamma_i(t)$ is the probability ...
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2answers
96 views

Markov chain probability in a complete graph

I'm having a hard time understanding the following. Assuming we have a complete graph with $n$ vertices, where initially, one of those has an information. At this point, all we care about is that in ...
3
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1answer
95 views

Probabilistic Substring Match

I'm looking for an algorithm that will help me determine substring matches at scale. I have a pool of 100+ million "needles" (strings). I can do as much pre-processing on them as I want, and storage ...
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1answer
36 views

“Learning occupancy grid maps with forward sensor models, S.Thurn,2008” a formula that is difficult to overcome

I am reviewing the document" Learning occupancy grid maps with forward sensor models, S.Thrun, 2008" (http://faculty.iiit.ac.in/~mkrishna/ThrunOccGrid.pdf) i am really confused about formula (26), ...
3
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0answers
58 views

Box labelling game

I have a box of stickers. It contains $n$ stickers. Each sticker is labelled with a different number from $\mathbb{Z}$. I have infinite supply of boxes. Box labelling game: I pick a random sticker ...
3
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2answers
89 views

Doesn't a quantum algorithm being deterministic contradict the superposition principle?

How can the EQP quantum complexity class exist? I mean, doesn't the superposition principle imply it is not possible to know with probability 1 whether a quantum algorithm will return a solution? For ...
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1answer
35 views

Recognizing vs Deciding in defining class BPP

In Sipser's text, he writes: When a probabilistic Turning machine recognizes a language, it must accept all strings in the language and reject all strings not in the language as usual, except ...
4
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1answer
194 views

Analysis of a randomized algorithm for independent set construction

Suppose that $G = (V,E)$ is a 3-regular graph on $n$ vertices and $m$ edges. Below I propose a randomized algorithm for obtaining an independent set for $G$. Step $1$: Delete each vertex (...
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1answer
68 views

Prove that we can change probability in definition of PP class

According to Wikipedia, PP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. If the answer ...
4
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1answer
323 views

Prove that PP is closed under complement

I found this proof in Wikipedia, but one of the most important steps in it doesn't make any sense to me. I'll explain: By the definition of PP there is a polynomial-time probabilistic algorithm ...