An algorithm whose behaviour is determined by its input and a source of random numbers.

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1answer
52 views

How to simulate this randomized “finite” sum? [closed]

Consider the the Geometric Brownian motion $\qquad dX_t=\mu X_t dt+\sigma X_t dW_t$, with $X_0=1$, $\mu=0.2$, and $\sigma=0.30$. for each $n=1,2,3,..$ let $h_n=1/n$, and let $X^n_n$ be the final ...
1
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1answer
28 views

How would you implement truly random hash functions in practice?

Suppose that $[U] = [0,...,U-1]$ is the universe from which all elements will be taken, and $A$ a hash table of size $m$. A hash function $h:[U]\rightarrow[m]$ is truly random if For any set of ...
3
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2answers
133 views

Minimal number of attempts at a multiple choice exam needed in order to pass, without any prior knowledge

A test is consisted of $N$ multiple choice questions, each has $k$ possible answers. A test solution is the sequence of answers $S\in[k]^N$. Given is a black box which receives a solution as input ...
0
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1answer
60 views

Random uniform sampling of position restricted permutations

Is there any efficient algorithm which is able to generate nearly uniform samples of permutations in case of position restrictions? Consider $N \times N$ restriction matrices $R$, that is matrices ...
2
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1answer
30 views

How to cluster similar objects into fixed size groups?

I have $n$ people each of which can meet on certain days of the week. I want to group them into $\frac{n}{k}$ groups of size $k$ such that all people in a group can meet on a day. eg - Suppose there ...
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1answer
27 views

Can a relatively small subset of random numbers be permuted and reused and still guarantee good expected running time for an algorithm like quicksort?

So this is sort of a general question but I'll limit the discussion to randomized quicksort to make it clear. Suppose generating "true" random bits is hard, e.g. because it requires measuring ...
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0answers
9 views

Removing the acceptance error from AM

Typically the AM class is defined with error upper bound of 1/3 for deciding both the situations of the membership question being true or false. But curiously enough for the situations when the ...
1
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1answer
24 views

How does derandomization of 3SAT work via conditional expectations?

Given a single SAT clause with its 3 literals coming from 3 different variables it is obvious that a random assignment of values will satisfy it with probability 7/8 But I do not understand how ...
3
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1answer
36 views

What is the rigorous definition of an efficient algorithm that $\epsilon-refutes$ random 3CNF formulas

I recently asked a similar What does "refuting random 3CNF" formulas mean?, however, I'd like to address it in a more mathematically precise setting. In that paper, on page 5, it talks ...
4
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0answers
55 views

Generate a random graph with geometrical degree distribution

I'm working on graph generation, trying to implement the RT-nested-Smallworld network model described in this paper. We are talking about generating an undirected graph in a slightly different way ...
2
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1answer
24 views

How does one change the probability bounds in probabilistic complexity classes without changing the class?

I see this theorem whose proof is not clear to me : "Let $L \subseteq \{0,1\}^*$ be a language and suppose that there exists a polynomial time PTM M such that for every $x \in \{0,1\}^*$ and $Pr[ ...
2
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1answer
27 views

Can one use the PCP theorem to prove correctness of deternimistic algorithms?

I am thinking of the equality "PCP(O(log(n)),0) = P" Say I have a deterministic polynomial time algorithm $A$ whose correctness I can't prove immediately. But say I create a probabilistic version of ...
3
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0answers
55 views

Analysis of sorting Algorithm with probably wrong comparator?

It is an interesting question from an Interview, I failed it. An array has $n$ different elements $[A_1, A_2, \ldots, A_n]$ (random order). We have a comparator $C$, but it has a probability p to ...
0
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1answer
31 views

Approximating the set of witnesses of a BPP algorithm

Let $\mathcal{A}$ be a randomized algorithm that decides a language $\mathcal{L}$. For each input $x\in\mathcal{L}$, we define the set of witnesses of $x$ as $W(\mathcal{A},x) = ...
4
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1answer
109 views

Amplifying the correctness of $\mathsf{RP}$ algorithms using expander graphs

A graph $G = (V, E)$ is called an $(n, d, \varepsilon)$-expander if the graph has $n$ vertices, maximum degree $d$, and satisfies the following expansion property: for every subset $W\subset V$ such ...
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2answers
71 views

About being able to sample a permutation of a finite set uniformly at random [closed]

I was looking at this question. So if I understand the above discussion right then it concludes that if say one had access to an oracle which can uniformly at random sample from a finite set then ...
0
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1answer
25 views

Complexity bound on $RP^{RP}$

This is a homework question, I'm wondering if anyone could help. Recall $RP$ is the set of languages recognized by randomized algorithms in polynomial time. The question is given an algorithm in ...
6
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1answer
78 views

Finding a maximal independent set in parallel

On a graph $G(V,E)$, we do the following process: Initially, all nodes in $V$ are uncolored. While there are uncolored nodes in $V$, each uncolored node does the following: Selects a random real ...
2
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0answers
93 views

Marking nodes of a complete binary tree

Suppose that I have a binary tree with $N = 2^h - 1$ nodes, initially all nodes are unmarked Over time via this process nodes became marked. Suppose that nodes have unique identifiers in range of ...
2
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1answer
30 views

Does randomness makes exponential difference?

Schwartz–Zippel lemma can solve the polynomial identity testing in expected poly-time. As far as I know, there is no deterministic poly-time algorithm for the problem, but we do not know if the ...
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1answer
80 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 ...
14
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1answer
292 views

Lost in a “one directional” concert

You and a friend lost each other on the line to a concert, and neither is sure which of you is further ahead. Formally, each is at some integer coordinate and may only walk towards a higher coordinate ...
2
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1answer
21 views

Using all the entropy in an biased bit

Suppose we have $n$ bits of random-looking data, and we want to encode it in such a way that instead of 1/2 the bits being 1's, we have (say) 3/4 the bits being 1's. The entropy of each bit in the new ...
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0answers
61 views

Proof of Randomized Self-Adjusting Binary Search Tree

I developed a randomized self-adjusting binary search tree years ago, which I called a shuffle tree, but was unable to ever have it published because my proofs were rejected (with little explanation). ...
2
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1answer
70 views

Turn biased random number generator into uniform

I'm looking at a problem in the book Introduction to Algorithms by Cormen et al. It says that if we are given a random number generator rand() which satisfies the distribution: $P(X = 0) = ...
0
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2answers
104 views

Genetic Algorithm Minimum Population Size

Is there a minimum limit to a pool (population) size when using the genetic algorithm to solve an optimization problem? For example a population of size 2.
1
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0answers
42 views

Tradoff between space and false positive rate when using bloom filters

Bloom Filters have false positive rate of $\epsilon = 2^{-k}$ with a data structure of size $m = n\log (\frac{1}{\epsilon})\ln 2$. Suppose you fix the number of hash functions at $k \le 3$. What is ...
5
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3answers
348 views

Constructing a random Hamiltonian Cycle (Secret Santa)

I was programming a little Secret Santa tool for my extended family's gift exchange. We had a few constraints: No recipients within the immediate family Nobody should get who they got last year The ...
3
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1answer
50 views

Asymptotic expected runtime of Randomized Algorithm

I am analyzing the asymptotic runtime of a randomized algorithm in expectation. The algorithm has the following properties: Given input size $n$, with probability $3/4$ it moves on to solve an ...
5
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1answer
76 views

Finding $k$ claws ($K_{1,3}$ bipartite graphs) in a graph?

Usually questions deal with claw-free graphs, but suppose we are given a graph $G$ and there are $k$ vertex-disjoing claws in the graph, how can we derive a randomised algorithm using color coding to ...
2
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1answer
162 views

Need a hint! Karger's algorithm versus Kruskal, spanning tree distribution

Let G = (V,E) be a unit-capacity graph with n vertices and m edges. Let T denote all the spanning trees in G. If we run Karger's algorithm, we will get a random spanning tree in T formed by the ...
2
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0answers
121 views

Message protocol to probabilistically infer missing object from Union of two subsets of a larger set

This was a challenge problem I read some time ago and just remembered it: Say you have two people, $A$ and $B$, collect objects distinctly labeled $1,...,n$. They will each separately collect ...
7
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1answer
505 views

Does a coin tossing algorithm terminate? [duplicate]

Suppose we have an algorithm like: n = 0 REPEAT c = randomInt(0,1) n = n + 1 UNTIL (c == 0) RETURN n (Assumuing the random number generator produces "good" ...
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1answer
30 views

Algorithm for random generation of two connected partitions of a finite set [closed]

Given the set $X=\{1,2,\ldots,n\}$; $\,\,$ $n=mp=kq$ where $m,k,p,q$ are positive integers. Please help me to programme an algorithm that realizes random generation of the following two partitions of ...
1
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1answer
31 views

Average Case runtime for random choice search

Assuming we have an array with $n$ Elements and want to find an unique element by randomly (uniformly) choosing. What would be the average case runtime? My thoughts so far: The chance to find the ...
3
votes
2answers
128 views

Are nondeterministic algorithm and randomized algorithms algorithms on a deterministic Turing machine?

An algorithm on an abstract machine is a finite sequence of operations of the machine. (Correct me if I am not correct.) However, there are different kind of algorithms, such as deterministic, ...
1
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1answer
42 views

Randomized agorithm for selecting a $m$-subset out of a $n$-set

The question has been discussed here but unfortunately is closed. Reference : m-element random sample being equally likely ...(CLRS 5.3-7)? My question is as follows. Suppose $n = 5, m = 3$. This ...
3
votes
1answer
37 views

Understanding the flaw in a proof attempt of the Communication Complexity of Equality

I'm new to communication theory and I've been wondering where the following simple argument fails: Equality Problem We have two players, player 1 Alice who gets an $n$-bit vector $X$ and player 2 Bob ...
1
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0answers
30 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 ...
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votes
2answers
111 views

Which type of randomized algorithm is best suited for web crawling?

I have decided to implement a web crawler for my CS major project. The project is focused towards adaptive search. I want the pages to be as user specific as possible and time efficiency is not much a ...
0
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0answers
90 views

Randomized algorithm to make a Binary Search Tree from an array of $n$ distinct elements

An array $\mathcal{A}$ of $n$ distinct integers $\{a_1,a_2,\ldots,a_n\}$ is given. I'm asked to design a randomized (esp. Las Vegas) algorithm to make a Binary Search Tree out of these elements, such ...
3
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0answers
42 views

Status of Research Problems in Motwani and Raghavan

One nice aspect of Motwani and Raghavan's classic textbook, Randomized Algorithms, is that the notes for many chapters include open questions marked as "research problems." However, the textbook is ...
12
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4answers
435 views

Simulate a fair die with a biased die

Given a biased $N$-sided die, how can a random number in the range $[1,N]$ be generated uniformly? The probability distribution of the die faces is not known, all that is known is that each face has a ...
5
votes
2answers
132 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 $k$ ...
1
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1answer
64 views

Name for class of algorithms preserving accuracy/confidence

I am considering the following class of algorithms: The algorithm has access to some probabilistic oracle (procedure) $f$ in addition to input. The answer of procedure $f$ (we may assume it is ...
1
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1answer
66 views

Understanding Property Testing with a toy example

I am newbie with this property testing and I am trying to understand it with a few examples. I first dealt with a toy example. I did not understand the first step of the test in the following slide. ...
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votes
1answer
175 views

m-element random sample being equally likely …(CLRS 5.3-7)? [closed]

I am trying to understand the following solution to CLRS 5.3-7: http://clrs.skanev.com/05/03/07.html Question description is on the page. I understood the part where m-element subset is constructed ...
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votes
1answer
55 views

Can someone explain LazySelect?

The LazySelect algorithm is given in these slides as follows. We have a set $S$ of $n = 2k$ distinct numbers and want to find the $k$th smallest element. Let $R$ be a set of $n^{3/4}$ elements ...
0
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2answers
71 views

Randomized Algorithm with matrices [closed]

We have two computers, Comp1 and Comp2, which hold binary matrices A and B of size $n\times n$. We want to check if the matrices of the computers are identical except for exactly 1 entry. Comp1 has ...
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votes
1answer
45 views

checking if there're equal bits in binary string [closed]

We have two binary strings, $X$ and $Y$, in two different computers. Both of them in length $n$. The computers can communicate by sending bits to each other. We have to build randomized algorithm to ...