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Questions tagged [probabilistic-algorithms]

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

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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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19 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 ...
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21 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 ...
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102 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 $Θ(N/ log(N))$. ...
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41 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 ...
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Trouble with one line of the proof of PAC learning axis-aligned rectangles

The proof for PAC learning axis-aligned rectangles usally goes like this : Let $\mathsf{A}$ be the algorithm that constructs the tighest rectangle $\mathsf{R_S}$ containing all positive samples seen ...
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39 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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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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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
104 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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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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32 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
45 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) ...
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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
47 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
38 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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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
291 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
35 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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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
33 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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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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37 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 ...
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1answer
98 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
41 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
180 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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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 ...
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1answer
81 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 ...
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157 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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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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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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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
121 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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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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109 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
34 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
54 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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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
75 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
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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 ...
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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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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 ...
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1answer
110 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
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“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), ...
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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 ...