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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2answers
34 views

Maximization problem

I work at a company and i got to a seminar we're they told us to solve this problem below in the picture Is there an algorithm that can help me solve this question. I thought about a randomized ...
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1answer
20 views

Randomly generating graph based off number of connections on each node

I'm trying to generate a graph based off some data I have. This graph should have N nodes where the number of edges each node has is equal to a random number ...
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1answer
22 views

Efficiently selecting a random subset of size $m$ from a set of size $n$

This is a cross post of my question here on math.se. I have a list of $n$ items and would like to randomly select an $m$ set from it efficiently (in terms of time complexity). Also, I want all ...
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22 views

Distributional error probability of deterministic algorithm implies error probability of randomized algorithm?

Consider some problem $P$ and let's assume we sample the problem instance u.a.r. from some set $I$. Let $p$ be a lower bound on the distributional error of a deterministic algorithm on $I$, i.e., ...
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1answer
30 views

Median of distribution with memory constraint

Task I want to approximate the median of a given distribution $D$ that I can sample from. A simple algorithm for this, using $n$ samples, is: ...
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3answers
187 views

Book recommendations on the analysis of randomized algorithms

I would like to read some books (or any other material) that cover the design of randomized algorithms with a particular focus on the analysis. My main goal is to develop the rigour needed to ...
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1answer
180 views

Weighted Online Matching - randomized algorithms

Let's consider the edge weighted online matching problem. The Vertices arrive online and reveal all their current edges and edge-weights $w_e>0$. The goal is to maximize the matchings weight. An ...
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0answers
22 views

Analyzing a counting triangles streaming algorithm which uses $\ell_0$ sampling

I'm trying to analyze the following streaming algorithm for counting triangles (see below). It supposedly works also for dynamic graphs (i.e. "turnstile model", where edge deletions are ...
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1answer
109 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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1answer
93 views

Randomized Algorithms: High-Probability vs. Expectation

Hopefully this question isn't too general, but I was wondering what the relationship is between randomized algorithms that perform well with high-probability and those that perform well in expectation....
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18 views

Space complexity of using a pairwise independent hash family

I'm trying to analyze the space complexity of using the coloring function $f$ which appears in "Colorful Triangle Counting and a MapReduce Implementation", Pagh and Tsourakakis, 2011, https:...
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1answer
71 views

Returning random integer from interval based on last result and a seed

Suppose we have an interval of integers [a, b]. I would like to have a function that returns random members from within the interval, without repetitions. Once that all members within the interval are ...
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1answer
39 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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1answer
36 views

1-sparse recovery algorithm

In the reference below, a 1-sparse recovery algorithm over a vector $a \in R^n$ is defined as follows. My question is why do we need the modulus (i.e. $x \mod p$)? Algorithm: Keep track of $$ \...
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1answer
25 views

Purpose of randomization/derandomization in basic randomized algorithm for MAX SAT

In Sections 5.1 of The Design of Approximation Algorithms by Williamson and Shmoys, they describe a basic randomized algorithm for MAX SAT and how to derandomize it. The algorithm is just to assign ...
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1answer
30 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
29 views

When does a Monte Carlo algorithm solve a problem?

When can we say that a Monte Carlo algorithm solves a problem? To quote from Wikipedia on Monte Carlo algorithms For instance, the Solovay–Strassen primality test is used to determine whether a ...
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Lectures and books for beginner to approach learning simulations

I'm an incoming undergrad with a math background up to single-variable calculus, but reasonably strong programming background through algorithms, data structures, web and mobile app development. ...
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1answer
49 views

Marginal Probability of Generating a Tree

Fix some finite graph $G = (V, E)$, and some vertex $x$. Suppose I generate a random sub-tree of $G$ of size $N$, containing $x$, as follows: Let $T_0 = \{ x \}$. For $0 < n \leqslant N$ i. Let ...
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0answers
39 views

How to estimate the number of elements inserted to a Bloom filter

A Bloom filter is a probabilistic data structure that allows encoding sets with false positives. Parameterized by the number of bits $m$ in the array $A$ (initialized to zeros), and number of hash ...
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1answer
32 views

Is there an efficient algorithm for determining the probability a large randomly chosen integer is not divisible by any integer of some set?

Given a set of 10 integers $A = a_1, a_2, \cdots a_{10}$, is there an efficient algorithm which can tell me what's the probability a randomly chosen integer between $1$ and $10^{10}$ is NOT divisible ...
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1answer
53 views

Flip-based algorithm for Delaunay triangulation in expected or average-case O(nlogn)

Focusing on the 2D plane: Lawson's Flip Algorithm works in worst-case $O(n^2)$ flips. I have seen it mentioned that (other?) flip-based algorithms work in expected $O(nlogn)$ time for two dimensions. ...
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0answers
29 views

Heuristic algorithm for the minimum weighted s-t cut with linear running time

To the best of my knowledge, the best algorithm for the minimum s-t cut in a weighted digraph is the Goldberg push-relabel algorithm with $O(n^{2}\sqrt{m})$ time complexity. I'm interested in solving ...
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1answer
108 views

Analyzing a randomized algorithm for finding an approximate median of an array

I'm given an array $A$ = ($a_1, a_2, \cdots a_n$), where n is uneven. For an element $a_i$ we denote its position in the array with $p(a_i)$. This element would be an $ε$-approximate median of the ...
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2answers
287 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
64 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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2answers
949 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
21 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
31 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 FPT ...
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1answer
48 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
38 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
545 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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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
174 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
39 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
36 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
66 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
397 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
48 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
30 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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25 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
83 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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139 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
39 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
56 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
638 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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0answers
29 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
59 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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1answer
66 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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59 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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