Questions tagged [randomized-algorithms]

Questions about algorithms whose behaviour is determined not only by its input but also by a source of random numbers.

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508 views

How can Karger's algorithm (and other randomized algorithms) be used in practice?

Suppose I am given the following problem (the source is here): Disconnect two nodes in a graph by removing minimum number of edges. I would apply Karger's min-cut algorithm. But how can I ...
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147 views

Describe a Monte Carlo algorithm for the Triangle Packing problem

Book: Parameterized Algorithms by Marek Cygan (free to download legally) Chapter about Multivariate polynomials on Page 353 (In the book not the pdf) Question 10.19: Describe a Monte Carlo $2^{3k}n^{O(...
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435 views

Can an algorithm be truly non-deterministic?

I read the term "non-deterministic algorithm" in many places but I don't see how an algorithm can be truly non-deterministic. Typically, there is some source of randomness in these algorithms. If the ...
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348 views

Karger's algorithm: why does every vertex have degree at least the number of edges crossing a min cut?

I'm currently watching a video on the analysis of Krager's Algorithm, and I am confused about something. The analysis goes as follows: Fix a min cut $(A,B)$. Let $k$ = # of edges crossing $(A,B)$ , ...
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87 views

Does randomness make 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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104 views

BPP upper bound

does $BPP\subseteq P^{NP}$ ? it seems reasonable but I don't know if there is a proof of this!could any one post a proof or any material that discusses the statement or something that look like this .
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86 views

Does this shuffle have non-zero probability for all permutations?

I was trying to do some code golf, when I created the following algorithm to shuffle a string: ...
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158 views

Why doesn't this put $BPP$ in $NP$?

From Sipser Gacs we know $x\in L(M)$ for a machine $M\in BPP$ $\iff$ $$\exists t_1,\dots,t_{|r|}\forall r\in\{0,1\}^{|r|}\vee_{i\in\{1,\dots,|r|\}}M(x,r\oplus t_i)=1.$$ From Adleman we know $x\in L(M)$...
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726 views

Compute median in unsorted array in $\mathcal{O}(\log{}n)$ space and $\mathcal{O}(\log{}n)$ passes

I want to compute the median in an array of size $m$ which consists of distinct integers from $\{0, 1, ..., n-1\}$, I have $m<n$. By median I mean the middle element (rounding up/down if the array ...
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2k views

What is a stateful computation?

I am reading about a specific field of probabilistic programming, and trying to understand what the term "stateful computation" means. See: http://projects.csail.mit.edu/church/wiki/...
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What is the trick used in skip lists to minimize $k + \frac{n}{k}$?

I was reviewing skip lists and the first step is to have two lists, the bottom one ($L_0$) of length n and the top one ($L_1$) of size k. Usually one traverses the "express line" (i.e. the top lane ...
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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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127 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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1k 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 ...
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123 views

Advantage of the Monte Carlo method over a regular periodic sampling [closed]

I am unclear on when to use the Monte Carlo random sampling method for algorithm design. The classic example that I keep seeing is using random points within some bounding rectangle to determine the ...
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45 views

Algorithm for cyclic $n$-string Hamming distance with constant sized language $\Sigma$

Suppose we are given a language $\Sigma$ where, suppose, $|\Sigma| = O(1)$. Consider two fixed strings $A, B \in \Sigma^n$. Define the Hamming metric between these strings as $$d_{H}(A,B) = \sum_{i=1}^...
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38 views

Generate a uniform random numbers in $O(1)$

Suppose you have access to a random number generator $G()$ that generates uniform random numbers in $\{0,\cdots,n-1\}$. (Here, $n$ is given and cannot be changed.) How do we generate a uniform random ...
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124 views

Is any randomized Algorithm a probability distribution over the set of deterministic Algorithms?

If there is a finite set of Instances of size n and the set of (reasonable) deterministic algorithms is finit. Can any randomized Algorithm be seen as a probability distribution over the set of ...
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44 views

Streaming algorithm for counting triangles in a graph

As described in the reference, the algorithm (see at the bottom) supposes to output an estimator $\hat T$ for the # of triangles in a given graph $G = (V, E)$, denoted $T$. It is written that "it ...
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380 views

Imperfection in randomness in VLC shuffle playlist - why?

Whenever I play a playlist of music using VLC (possibly other software too), I notice that some songs never get played while others get played repeatedly (even for a playlist of just 8 songs). I know ...
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187 views

Generate random matrix and its inverse

I want to randomly generate a pair of invertible matrices $A,B$ that are inverses of each other. In other words, I want to sample uniformly at random from the set of pairs $A,B$ of matrices such that ...
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95 views

Measuring the Probability of Error for a Potential BPP Algorithm

Problem Given a search algorithm that can be used to query a k-dimensional space, produced from an input array of N data, has a time complexity of $O(klog^2N)$. This algorithm partitions the space ...
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Average case of simple algorithm like binary search

These questions is about one of my research. As I am not a computer scientist, formal answering is difficult to me. I have a special search algorithm which the explanation here will take a lot of ...
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568 views

Generating uniformly random bits from a stream of arbitrarily biased bits

Say we have a function called GenBiasedBit. This function returns 1 with probability p (where p is an unknown real number between 0 and 1 exclusive) and returns 0 with probability 1 − p. How could I ...
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TSP problem with a benchmark data

I've got a test Travel Salesman Problem's data with known optimal solutions. It's in a form of set of 2D points. Particularly, this is a tsplib format; sources are here and here. I'd started a ...
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67 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) = \{r\in\{0,1\}^n:\...
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256 views

Guessing the best choice to maximize returns

There are $N$ number of people and $X$ amount of objects with different values. Each person will choose an object and will obtain that objects value. If multiple people choose the same object then the ...
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45 views

What is the advantage of probability algorithm?

What is the advantage of probability algorithm? e.g. Las Vegas. I would also like to know some applications of the randomized algorithms. Are there any recommendable courses or books?
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Quicksort: Probability of an element being compared to fewer than $k$ pivot elements

Assume we want to use quicksort on some array $s$ with length $n$ consisting of only $n$ distinct elements. Let $S_{(1)},S_{(2)},\dots,S_{(n)}$ be the sorted order of the elements in $S$. Furthermore, ...
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1answer
50 views

Set which is easy to sample, but difficult to sample from its complement

Given a set $S \subseteq \{0,1\}^*$, the algorithm $A$ is a generator for $S$ if given $n$ random bits $x \in \{0,1\}^n$, $A$ generates an element of $S$ of size $n$, and $A$ can generate at least $\...
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364 views

Introduction to Algorithms (CLRS) Ex 5.2-5 solution

The following is Ex 5.2-5 from Introduction to Algorithms (CLRS), 2nd Edition. Let $A[1...n]$ be an array of n distinct numbers. If $i<j$ and $A[i]>A[j]$, then the pair $(i, j)$ is called an ...
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437 views

How can maximum number of minimum cuts of a graph be exactly $n \choose 2$?

According to my instructor, $n\choose 2$ is the maximum number of minimum cuts we can have on a graph. To prove this, he showed the lower bound using an n-cycle graph. To prove the upper bound, he ...
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65 views

Question about Morris' algorithm

I am reading the lecture notes. I am trying to understand Morris' algorithm on page 2. The Morris' algorithm is as follows. Problem: Given an input stream $\sigma$, compute (or approximate) its ...
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1answer
199 views

Weighted probability using Huffman Tree

I want to produce a value from a set, where each value has an associated weight. Eg: [(1, 4), (2, 3), (3, 3)] should give me a 40% chance of picking 1, and a 30%...
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1answer
63 views

Choosing $n$ points to get the closest sum

Let's say there is a set of $N$ real numbers, $x_i, i\in\{1,2,...,N\}$, and we would like to choose $n$ points out of them to get the sum of the chosen points as close as possible to a certain number, ...
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1answer
256 views

detecting a cycle in an undirected graph problem is in $RL$ complexity class

I need come up with an algorithm for detecting a cycle in an undirected graph where the algorithm is in $RL$. That is, the algorithm detects a cycle with a probability greater-equal to $\frac{1}{2}$ ...
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1answer
123 views

What is the Expected run time of an algorithm that randomly generates N unique strings of length D?

I was trying to rigorously/mathematically analyze the runtime of the following algorithm: ...
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1answer
65 views

Perturbing trees

I have a collection of labelled directed trees, and from these input trees I would like to generate permuted trees that have the same node set but whose edges and labels have been permuted with some ...
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1answer
415 views

Methods for proving upper bound on a-approximiation algorithms? [closed]

I'm dealing with some simple randomized and on-line algorithms, both kind produce some lower/upper bound on quality of the output instance. For example, there's a simple randomized algorithm for the ...
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1answer
137 views

Deamortizing a Las-Vegas randomized algorithm

Deamortization refers to the process of converting an algorithm with an amortized bound into one with a worst-case bound. For example, assuming you need to find the median of an array once every $n$ ...
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666 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 from ...
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68 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[ M(...
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57 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 ...
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741 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, non-...
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173 views

Interpretation of “expected cost” of an algorithm

I'm studying randomized algorithms and I sometimes come across results like (1) The algorithm has an expected $O(f(n))$ cost. and (2) With constant probability, the cost is bounded by $O(f(n))$....
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1answer
55 views

2-Approximation algorithm for for messages across a cyclic network

Question There are $n$ computers arranged in a cycle ($1,2,3..,n,1$), with undirected edges between adjacent computers. There are $m$ messages that need to be delivered. Message $i$ ($1 \le i \le m$) ...
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1answer
45 views

Confusion about the Hiring Problem

I'm confused about where the probability from the hiring problem comes from. For background: We interview one person everyday who has a quality characteristic, x, from 0 to 1(distributed uniformly). ...
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1answer
40 views

If I walk through list and delete every out-of-order element I come across, on average how many elements will be left?

I have a uniformly randomly permuted list of length $n$. I walk through the list element-by-element, and delete an element if it's out-of-order (compared to the previous in-order elements of the list)....
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1answer
91 views

Network throughput with random delay selected from uniform distribution

Background: I am working with IoT devices which broadcast status messages over a wireless channel periodically and at a rather high rate (500-5000 Hz). Receiving every message is not crucial but the ...
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1answer
42 views

Sublinear Homomorphism Property Testing Counter Example

This is a homework question, so I'm not looking for answers, just general guidance. I'm looking at a Sublinear Algorithms survey where (Group) Homomorphism property testing is discussed. The case of ...

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