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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43 views

Probability that two elements are compared in randomized quicksort

I am having an issue in a specific part of the randomized quick-sort analysis. As per the randomized quick-sort algorithm the pivot is chosen from the given subset on which it is called from a random ...
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1answer
50 views

Amount of expected loop iterations when searching an array by random index

Lets say we have an array A of size n. It has 1 as its first index and n as its last index. It contains a value x, with x occurring k times in A where 1<=k<=n If we have a search algorithm like ...
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909 views

Generating random words by grammar

A bit of context I was writing a parser for a grammar, and for testing purposes I come up with idea to generate some random inputs. The grammar I was dealing with was much more complicated, in this ...
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0answers
18 views

An algorithm which efficiently generates random samples without replacement, from a large range [0-N], N ~ 10^12?

I want an algorithm which generates random integers, without replacement, from a large range [0-N], N~10^12. However, the whole range should not be stored in memory. The memory footprint should be O(...
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1answer
23 views

Color coding to get an FPT algoirthm for $k$ disjoint triangles

Consider the following problem: Input: A graph $G=(V,E)$ and an integer $k \in \mathbb{N}$ Output: Are there $k$ vertex-disjoint triangles in $G$? Assume we want to use color coding to develop an ...
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1answer
47 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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1answer
37 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
449 views

Expected length of a random walk on a line

I am given the following randomized algorithm for SAT, Input: A satisfiable CNF-formula $\varphi$. Output: An assignment $\rho$, such that $\rho \models \varphi$. The algorithm works as follow: ...
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114 views

Compute the expected size of an approximation of vertex cover

Consider the following randomized approximation algorithm of vertex cover: Input: A graph G = (V, E). Output: A set $C_G \subseteq V$ a vertex cover of $G$. The algorithm: Set $C_G := \emptyset$. ...
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1answer
167 views

Error lower-bound for an algorithm for vertex cover

I have the following randomized-algorithm for the vertex cover problem. Let $B_0$ be the output set: Fix some order $e_1, e_2, . . . , e_m$ over all edges in the edge set E of G, and set $B_0 = \...
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1answer
30 views

Given an unsorted list of $n$ items, how many random comparisons are needed on average to be able to sort the list?

There is an unsorted list of $n$ items $x_1, \ldots, x_n$. Until you can sort the list, you are given one of the ${n \choose 2}$ possible binary comparisons uniformly at random (with replacement). On ...
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0answers
31 views

Finding a node in a binary tree by looking at the path between it and the root

There is a directed binary tree as shown in the picture (all edges are diercted from higher- to lower-level nodes). In that tree there is some specific unknown node $s$. All nodes in the $(s, root)$ ...
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1answer
36 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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4answers
388 views

How to devise an algorithm to generate a random but valid train track layout?

I am wondering if I have quantity C of curved tracks and quantity S of straight tracks, how I could devise an algorithm, (computer assisted or not), to design a "random" layout using all of those ...
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1answer
43 views

Doubt on Karger's Algorithm for Min-Cuts

I am learning Karger's algorithm for Min-Cuts.I have been solving 1 problem on it. The first part of the problem asks us to run Karger's Algorithm on a given graph. I have no problem doing that. My ...
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0answers
26 views

Randomized version of the class $APX$?

Is there a class which is to APX what BPP is to P? I'm looking for a definition that is like the following: "For $r > 0$, an $r$-RPCA (randomized polynomial-time constant-factor approximation) ...
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24 views

Bayes theorem and randomized algorithms

Are there any randomized algorithms that make use of Bayes theorem? where are they used and why?
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1answer
79 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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99 views

What is the exact time complexity of randomized Kuhn's algorithm?

Please, read the whole question before answering, the exact details of the implementation are important. Suppose that you want to find largest cardinality bipartite matching in bipartite graph with $...
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0answers
37 views

Karger's min-cut (contraction): Combinatorial argument for success probability?

The contraction algorithm for min-cut is: pick an edge $(u,v)$ uniformly at random, and "contract" it by merging $u$ and $v$ into a single vertex, deleting self-loops. Continue until two vertices ...
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1answer
40 views

What is the probability of comparision between smallest and greatest element in array when quick sort randomly choose the pivot element?

Consider the recursive quick sort with random pivoting i.e. each time a random pivot element is chosen uniformly. When this ...
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1answer
620 views

Random restarts for unsatisfiable instances

In the worst case, Boolean satisfiability (assuming P!=NP) takes exponential time. Nonetheless, modern SAT solvers using variants of DPLL, are able to solve enough instances to be useful in practice. ...
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9 views

Algorithm for Autonomously Culling Swarm

I'm trying to find an algorithm that would be able to cull a swarm of unknown size to around a known number with no overarching controller or dynamic registry (Each node should be able to decide ...
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0answers
28 views

Random Teams based on Positions in Sports

I have tried to find an answer to this but haven't found exactly what I'm looking for. I am trying to develop a way in which I can have a random team selected based on skill set (1-5 with 5 being ...
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1answer
55 views

Finding efficient randomized algorithm

I'm doing a course on randomized algorithms and I've encountered a question that I'm struggling to solve. Given a system of $m$ linear equations with $n$ variables over finite field $\mathbb{F_2}$ ...
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16 views

Algorithmic question: distribute balls, optimise for balancing (i) weights (ii) probabilities of picking balls

I have an algorithmic problem that requires some lengthy explanation, which follows below. tl;dr: distribute balls with weights among bags, optimise for balancing both (i) the weights between the ...
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0answers
28 views

CNN Predicting One Class and Accuracy Getting Stuck

My model is a binary classifier. With the same exact architecture, the model sometimes gets high accuracy (90% etc), other times it predicts only one class (so accuracy is stuck at one number the ...
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1answer
50 views

Why is $ZPP \geq BPP$ not true?

This seems like a silly question, but I couldn't find a conclusive answer for it. As far as I know, ZPP contains algorithms which run in polynomial time and either return a known-correct answer or ...
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57 views

Alternative criterion for approximate maximum-weight perfect matching algorithms [closed]

Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not the factor between the approximate and exact weight sum achieved by each solution, but ...
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1answer
112 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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1answer
112 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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1answer
29 views

Questions about Randomized Median algorithm?

In textbook by Mitzenmacher and Upfal here, they write in page 62, the following: By repeating Algorithm 3.1 until it succeeds in finding the median, we can obtain an iterative algorithm that never ...
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2answers
106 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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1answer
78 views

Can BPP be bounded around any constant other than 1/2?

A language $L$ is in BPP if there exists a randomised TM such that it outputs a correct answer with probability at least $1/2+1/p(n)$ for some polynomial $p(n)$, where $n$ is the length of the input. ...
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0answers
21 views

Random paths from one point to another going through all the cells of a square grid

I am looking for a very specific algorithm, so I think it doesn't exist yet. I would be satisfied if anyone was able to give me some hints to develop it. My problem is about a square grid of size <...
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0answers
34 views

Correctness of Karger's min-cut Algorithm

tl;dr in the analysis for Karger's min-cut, the probability of an edge being in the min-cut in the $j$th iteration, $\frac{k}{0.5k(n-j)}$, neglects the fact that all the edges between the two ...
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1answer
32 views

Non-existence of approximation algorithm for the knapsack problem

I am working on the following exercise: Prove that if $P \neq NP$, there cannot exist an approximation algorithm $A$ for the knapsack problem (KP) such that $\exists k \in \mathbb{N}, \forall I \in S: ...
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0answers
56 views

Randomized algorithm to compute cover radius?

I am self-study the book "Geometric Approximation Algorithms" by Sariel Har-Peled. And I stuck on a problem and don't know how to start it. Let $C$ and $P$ be two sets of point in the plane , such ...
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1answer
44 views

How to approach analysis of randomized algorithm

Let us suppose we have a sequence of values $C(i)$ that represent some counter for a given $i$ for $i \in \lbrace 1, \cdots, n \rbrace$. Let us assume some uniform distribution $U$ where selecting any ...
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0answers
12 views

Randomized response vs Output Perturbation

what are the difference between Randomized response and output perturbation? The only one I can think of is that Randomized Response's output is not always perturbated while in the other case it is.
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1answer
58 views

Using Chebyshev to derive an upper bound for Coupon Collecter's Problem

I'm TA'ing a course and have trouble solving an exercise. Let $X$ be a RV defined to be the number of trials required to collect at least one of each type of coupon (of which there are $n$). Then $E[...
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0answers
18 views

Min Cut Algorithm using Randomly inserted directions

I had a question about a different randomized min cut algorithm (I don't think it is as efficient as Karger's algorithm for larger sizes of min cuts but it is more efficient for smaller ones). My ...
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1answer
68 views

Prove the probability of which a hash function is collision-free

Suppose $H = \{h_1, ..., h_T\}$ be a family of pairwise independent hash functions mapping $\{0, 1\}^n$ to $\{0, 1\}^{n/2}$. Let $M = \frac{2^{n/4}}{10}$ and let $x_1, ..., x_M$ be any M distinct ...
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1answer
78 views

Constructing hitting sets for randomized algorithms

Suppose A($\cdot$,$\cdot$) is an efficient randomized algorithm and L is a language such that $\text{If }x \in L, \text{Pr}_r[(A(x,r) = 1)] = 1$ and if $x \notin L, \text{Pr}_r[A(x, r) = 0] \ge \...
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1answer
58 views

What properties does a co-RP problem need to be in P?

Given an arbitrary language $L$ with an algorithm $A$ that places $L$ in $co-RP$ what other properties does the Algorithm $A$ need to have to so that $L$ is in $P$? For example: Considering $L$ is ...
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6answers
8k views

Can we generate random numbers using irrational numbers like π and e?

Irrational numbers like $\pi$, $e$ and $\sqrt{2}$ have a unique and non-repeating sequence after the decimal point. If we extract the $n$-th digit from such numbers (where $n$ is the number of times ...
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0answers
102 views

How to generate random strings from Context-Free Grammar in GNF

I need to generate random strings given a grammar in Greibach Normal Form. The naive approach would be to generate a random integer n and perform ...
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0answers
34 views

Difference of simulated annealing and random search for generating crossword puzzles?

I heard that when one wants to write a program to make a crossword puzzle, he can use for example simulated annealing as in the thesis Crossword Construction using Constraint Satisfaction and ...
2
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1answer
87 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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2answers
63 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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