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

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2
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1answer
38 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. ...
2
votes
1answer
20 views

2D random walk. Should both dimensions be independent?

My assignment is to compare several probability distributions in random walk algorithm. I'd like to analyse it in 2D linear space to make the concept more intuitive. What is the correct approach in ...
2
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1answer
29 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 ...
5
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2answers
97 views

Is this special case of a scheduling problem solvable in linear time?

Alice, a student, has a lot of homework over the next weeks. Each item of homework takes her exactly one day. Each item also has a deadline, and a negative impact on her grades (assume a real number,...
2
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2answers
58 views

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 ...
1
vote
1answer
39 views

How can simulated annealing be related to the vehicle routing problem?

I have been searching through internet how could simulated annealing to solve the vehicle routing problem, but didn't find anything that made it clear to me. Most of what I found are research papers ...
3
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3answers
400 views

Are there adversarial inputs for randomized quicksort?

Someone recently claimed that there's an adversarial input for randomized quicksort; he referenced this paper. This defies my intuition because there are results that say that randomized quicksort ...
1
vote
1answer
24 views

Is this a kind of “sketching”?

Say one is given a matrix (assume real and symmetric if necessary) and its $n-$dimensional columns be say $v_1,v_2,..,v_n$. Now is it possible to find a set of $d<n$ lower dimensional vectors ($w_1,...
5
votes
1answer
56 views

Can we derandomize subexponential algorithms given P=BPP?

Under $BPP=P$ conjecture randomization does not have much power for poly time algorithms. Can we say the same about randomized subexp algorithms like number field sieve?
1
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1answer
51 views

Why is ZPP = RP ∩ co-RP?

I am trying to prove the theorem that ZPP = RP $\; \cap \; co-RP$. If $L \in \; \subseteq RP \; \cap \; co-RP$ then I can see that it belongs to $ZPP$. But I am unable to prove the reverse direction, ...
0
votes
1answer
46 views

How to compute Jacobi symbol efficiently?

How do I compute the Jacobi symbol $(N|A)$ efficiently? In particular, for every odd $N, A$, define the Jacobi symbol $(A|N)$ as $\prod_i Q_{p_i}(A)$ where $p_1, \dots , p_k$ are all the (not ...
3
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1answer
52 views

A clarification on $PP$

Wiki in https://en.wikipedia.org/wiki/PP_(complexity) says "a PP algorithm is permitted to do something like the following: On a YES instance, output YES with probability $1/2 + 1/2^n$, where n is ...
4
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2answers
63 views

On certificates in BPP (avoiding majority vote)

Assume that we have a $BPP$ algorithm $A$ for a problem $\Pi$. Given input $x$ we run $A$ on $\Pi$ polynomially many times and take majority output. However if the problem $\Pi$ is also in $NP$ ...
9
votes
1answer
63 views

Randomized Meldable Heap - Expected Height

Randomized Meldable Heaps have an operation "meld", which we then use to define all other operations, including insert. The question is, what is an expected height of that tree with $n$ nodes? ...
1
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0answers
50 views

How can we get a Las Vegas algorithm from a Monte Carlo one?

I am trying to solve some exercises on random algorithms from this book, randomized algorithms. This is not a homework. I am only trying to improve my skills. Here is the exercise: Exercise 1.3: ...
6
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1answer
145 views

What is the approximation ratio of this randomized algorithm for finding matchings?

I would like to analyze the following algorithm in terms of its approximation ratio. Here is the algorithm: ...
3
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1answer
67 views

What does the “principle of deferred decisions” formally mean

I have encountered the phrase "Principle of deferred decisions" in Mitzenmacher and Upfal's book on Randomized Algorithms and several other courses online. Isn't it just conditional probability? In my ...
2
votes
0answers
24 views

min cut for multiple partitions

So I am familiar with the standard minimum cut problem in which the goal is to find the smallest possible set of edges in a graph such that, upon their removal, we have two nonempty, disjoint ...
2
votes
1answer
53 views

Does hashing under the Simple Uniform Hashing Assumption battle worst-case adversaries the same way quick sort does?

One common way for algorithms to battle adversarial inputs is by acting randomly. One popular example is quicksort and choosing pivots randomly (this sort of notions is explained well in section 5.3 ...
0
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1answer
43 views

On deterministic weighted graph isomorphism from randomized

Is there a $O(n^2)$ algorithm to resolve isomorphism between two weighted $n$-vertex graphs? This is a much easier problem than graph isomorphism. Basically take an real edge weight set $\{w_1,\dots,...
1
vote
1answer
70 views

How do multiple branches from one node occur with the Monte Carlo Tree Search?

I think I understand the Monte Carlo Tree search. It goes through the tree until it reaches a leaf node, where it branches (creates a child node). However, the branching only occurs at the leaf nodes (...
3
votes
2answers
61 views

Are Random Forests Turing-Complete?

Is the random forests algorithm Turing-complete? As in, can any algorithm be represented by a given "tree" in the forest?
1
vote
1answer
112 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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0answers
14 views

What use does the predictor stage in a particle filter have?

Im a bit confused about the particle filter. I understand the generic particle filter algorithm but in some literature say particle filter has predictor stage which is not mentioned in the generic ...
8
votes
3answers
282 views

Isn't polynomial identity testing over arithmetic *expressions* trivial?

Polynomial identity testing is the standard example of a problem known to be in co-RP but not known to be in P. Over arithmetic circuits, it does indeed seem hard, since the degree of the polynomial ...
2
votes
1answer
38 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 ...
6
votes
3answers
129 views

Is there a random shuffle algorithm using only true /false?

Is there a way to randomly shuffle an array using only a source of random boolean values? SO to clarify, shuffle using true /false only, and not integers or decimals. For this question, I'm ...
3
votes
2answers
92 views

Shuffling a file on disk using $O(\log n)$ memory

How do you shuffle the bytes in a file (bytes for simplicity) on disk with a small, $O(\log n)$, amount of memory and preferably in-place? If the file had size $2^m$, then we can first split the file ...
2
votes
2answers
93 views

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 ...
1
vote
2answers
62 views

Is shuffling a set of items after popping an item meaningfully more random than doing it once, before starting?

I'm working on a thing to randomly assign people into a shift. There's mostly 2 sets of people, "free" and "assigned". Is shuffling the "free" set after assigning an employee meaningfully more random ...
8
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2answers
131 views

Is there any efficient algorithm for primality testing for numbers that are of the form $4k+3$ using the square root function?

I was reading CLRS and it asked to show that if $p$ is a prime of the form $4k+3$ and $a$ was a quadratic residue, then $a^{k+1}$ is a square root (one can also easily show that $a^{-k}$ is a square ...
3
votes
1answer
68 views

Optimal pivot selection for quick-sort

The actual runtime of applying quick-sort to an integer array heavily relies on the choice of pivots. It is well known that picking a random pivot does not work as good as taking the median of three, ...
3
votes
1answer
31 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 $n$ ...
4
votes
0answers
154 views

Data structures for ordering noisy data

In a certain robotics application, I encountered a problem in which we need to determine the order of positions of several robots on $\mathbb{R}$. Each measurement that we take of robot positions is ...
0
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0answers
14 views

constant step-size assumption in non-asymptotic analysis of stochastic gradient descent

I have come across some non-asymptotic analysis of stochastic gradient descent algorithm (works of FRancis Bach https://hal.archives-ouvertes.fr/file/index/docid/831977/filename/newsto_hal.pdf) most ...
6
votes
2answers
91 views

Correctness of Freivald algorithm for checking matrix multiplication, why is the probability of checking $AB \neq C$ at least 1/2?

I am going to consider Freivald's algorithm in the field mod 2. So in this algorithm we want to check wether $$AB = C$$ and be correct with high probability. The algorithm choose a random $r$ n-...
1
vote
1answer
81 views

How does one compute the probability of a false-based Monte Carlo algorithm being correct rigorously?

Recall that a false-biased Monte Carlo (MC) algorithm is always correct when it returns false for some decision problem i.e. it has a one sided error and its always correct on NO instances. Assume ...
4
votes
1answer
181 views

Choosing a random bit from a bitmap

Since, I don't have strong algorithmic background my question may sound a litlle odd. Please correct me, if so. I have quite a large bitmap (~100 Million bits) (e.g. ...
6
votes
1answer
109 views

Sorting an unordered pile of items into drawers with minimal drawer movements

A while ago, I was doing my laundry late at night. When I brought my laundry back to my dorm, I started to put it away. My wardrobe is set up as follows: My drawers are categorized by the type of ...
0
votes
1answer
190 views

Showing that Karger's contraction algorithm has exponentially small probability of finding an optimum

I'm stuck with one of my homework exercises: Consider the following variant of Karger’s algorithm for finding a minimum s-t cut, i.e., a minimum cut separating two specific given nodes s and t: ...
2
votes
0answers
45 views

Using the random forest algorithm to predict vectors [duplicate]

I know this might sound like a newbie question, but bear with me. I have read a paper where researchers use a random forest to predict species distribution, but in their study, they only predict a ...
1
vote
1answer
41 views

Proving simple bound on coupon collector

I came across this paper which gives bounds on coupon colloector problem. Page 451 contains a table where reference to U1 is given as 'folklore'. I presume this is trivial to follow from the ...
6
votes
1answer
72 views

Why is the probability used in the definition of RP complexity classes, arbitrary?

I was looking at the following wikipedia article on the RP complexity class: https://en.wikipedia.org/wiki/RP_(complexity) In its definition it states: If the correct answer is NO then it always ...
1
vote
1answer
72 views

Finding the kth smallest element of an array using DAC [closed]

I'm trying to find the $k^{th}$ smallest element of an array by using Randomized Quicksort, But below code giving erroneous result can anybody help. ...
0
votes
0answers
9 views

Find all pairs of matching elements in two arrays [duplicate]

I have two arrays A & B, each with an element from one that corresponds to the other (for each a in A there is a b in B it corresponds to). And I don't want to brute-force check them all in O(n^2)....
0
votes
1answer
45 views

How to “properly” handle incorrect values in a random walk?

I'm performing a simple one dimensional walk to create sample interest rates. Whilst I know there are lots of options for encouraging values to oscillate around a mean etc. I'm yet to find a simple "...
1
vote
1answer
58 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
vote
1answer
85 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 ...
4
votes
2answers
180 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
75 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 ...