Questions about (typically randomized) algorithms that can produce no or an incorrect answer with a certain probability.
4
votes
2answers
45 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
28 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
29 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
35 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
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}{|...
1
vote
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 ...
1
vote
1answer
33 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
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 ...
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,...
0
votes
2answers
151 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
27 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
0answers
18 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 ...
0
votes
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).
...
0
votes
1answer
24 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
vote
0answers
28 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?
0
votes
0answers
32 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
votes
1answer
84 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
31 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
47 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 $n > N_c$,
$$\...
3
votes
1answer
115 views
Algorithm for finding cliques
Given an arbitrary undirected graph $G = (V,E)$, I am interested in a low-polynomial time algorithm which can find several moderately large (ideally $O(n^\epsilon)$ vertices per clique for $\epsilon &...
0
votes
0answers
10 views
Can the following be viewed as a separation of complexity classes?
Suppose I'm trying to simulate a coin flip with probability $p$ of landing heads. But I'm only given access to perfectly random bits where each outcome is equally likely. If $p$ is a nice number like $...
2
votes
2answers
33 views
Cuckoo filters for non powers-of-2
The Cuckoo filters paper (https://www.cs.cmu.edu/~dga/papers/cuckoo-conext2014.pdf) claims a 95% load factor, however it seems to make an implicit assumption that the table size is a power of 2, and ...
2
votes
1answer
70 views
Confuse on using bucket number to replace hashing multiple times in hyperloglog explanation
In this blog the author states that the following
So we now have a rather poor estimate of the number of values in the
dataset based on bit patterns. How can we improve on it? One
...
2
votes
1answer
153 views
Proving $P=BPP$ under a assumption
Let suppose that there exists deterministic polynomial algorithm $A$ that approximates the probability that a given boolean circuit $C$ accepts random input with an error at most $\frac{2}{5}$. More ...
-1
votes
1answer
99 views
Graph isomorphism in BPP implies it is also in RP
$$L=\{\langle G\rangle \#\langle H\rangle : H, G \text{ are directed isomorphic graphs }\}$$
$\langle G\rangle$ is adjacency matrix written row by row.
Show that if $L\in BPP$ then also $L\in RP$.
Can ...
0
votes
0answers
35 views
Show that if language belongs to $BPP$ then also similar language belgons to $QP$
Prove that for each arbitrary set of natural numbers $A$ we have:
$$\{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 \...
1
vote
0answers
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?
2
votes
1answer
99 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:
...
0
votes
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 ...
1
vote
1answer
102 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 ...
1
vote
1answer
31 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 ...
1
vote
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/...
1
vote
0answers
48 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 ...
1
vote
1answer
58 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 ...
1
vote
1answer
28 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
votes
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 ...
2
votes
2answers
93 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
votes
1answer
88 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 ...
1
vote
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
votes
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
votes
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 ...
1
vote
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
votes
1answer
179 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 (...
-1
votes
1answer
67 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
votes
1answer
296 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 ...
0
votes
0answers
73 views
Algorithm to sort elements of an array under uncertainty
I have an array of $n$ elements. Each element is basically the mean of gamma distribution and the actual value of the element is picked from that distribution. Assume the parameters of distribution (...
2
votes
1answer
163 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. ...
1
vote
0answers
66 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
196 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
390 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 $\...
