Mathematical questions about Algorithms, including the analysis of algorithms, combinatorial algorithms, proofs of correctness, invariants, and semantic analyses. See also (comptutational-mathematics) and (computational-complexity).

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How to use mobius-inversion to solve this problem?

Currently, I'm trying to solve this problem using mobius-inversion. the function f(d) means the number of (i, j, k) equals d, and function g(d) means the numbers that satisfying: d | (i, j, k). Then ...
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1answer
40 views

Given that Minimum Spanning Tree is NP-complete show that Hamiltonian Cycle is NP complete

So first of all I know finding MST is in P and is not NP complete. But I checked last year exams from my University and there is a problem: Given that Minimum Spanning Tree is NP-complete show that ...
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17 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
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1answer
23 views

How to efficiently list prime with a very specific property

I noticed that my phone number 06 xx xx xx xx is a prime number. Ok that cool ... But if you had the country code (+33 for france), ...
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8 views

Is the mapping achieved by the discrete Frechet Distance the best achievable pair decomposition?

I have the following proble: There are 2 sequences $P$ and $Q$ composed by $M$ and $N$ points respectively. I want to calculate the discrete Frechet distance between them. This operation outputs also ...
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17 views

A suitable algorithm to change strings to zero strings

I found a programming exercise I couldn't solve. Namely, I have been given a long binary string and a fixed set of operations. Here, an operation means that it takes a substring and transforms its ...
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22 views

Regular Subresultant PRS from Euclidean PRS?

Is there any way to compute regular subresultant polynomial remainder sequence if we know the Euclidean polynomial remainder sequence of two univariate polynomials and vice versa?
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0answers
37 views

How many counts the square number?

For example I have a number is $55$. Can I check this for how many counts number's square create this number. For example I have $55$ So, I know this number total from 5 number, which are $$55 = ...
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1answer
22 views

Why is $X_m$ and $Y_m$ not included in the shaded region(where median can lie)?

This problem is from Algorithms, problem 2 The Problem Given two sorted list of numbers $X$[1..$n$] and $Y$[1..n]. we need to come up with a O($\log n$) time algorithm to find the median of the 2$n$ ...
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21 views

Help understanding Jacobi SVD

I found this link, and I want to complete this implementation of the Jacobi SVD method, but it isn't clear to me how to implement alpha, beta, and gamma. I think it's very clear that ...
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0answers
9 views

Code that Creates a Monotonic Array

I'm not sure whether this belongs on StackOverflow or here. Suppose I have an infinite (indexed from $-\infty$ to $\infty$) array of random doubles x[]. They are ...
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54 views

Squaring a matrix using a linear memory

I have a N x N matrix (let's denote it with A). I want to calculate $A ^ 2$, using $\theta(N)$ memory (speed does not matter as long as it's a polynomial) on one processor. I believe that this can ...
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1answer
53 views

What algorithm can sort the first sqrt(n) elements of an array in O(n) time?

I want to partially sort an array of $n$ elements to get the first $\sqrt{n}$ elements sorted, and it has to be done in $O(n)$ time. The complexity $O(n)$ seems to imply that it is necessary to go ...
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1answer
21 views

Finding algorithm to design combinatorics formulae per input data

Problem: calculate how many unique "words" can be made using the letters of your a) name (Kateryna - 8 letters, letter 'a' used twice) b) lastname (Atamanchuk - 10 letters, letter 'a' used 3 times) ...
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0answers
16 views

Computing efficiently a small base to the power a large number

Is there a fast algorithm to compute an exponential with a small base, (namely , close to 1) For example, computing 1.01 to the power 100?
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12 views

Depths of top-level multiplication algorithms

I've seen that the depth of the Cantor/Kaltofen algorithm is in $O(\log n)$. Are the operations for this complexity undifferentiated ? Or this complexity is in terms of multiplications only ?
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7 views

BFS Modification For Total Shortest Paths

I was given the following problem as an assignment but it is really confusing me: Consider the BFS algorithm. Given a digraph G = (V, E) and a starting vertex s ∈ V, this algorithm computes for ...
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1answer
45 views

The output spanning tree of Kruskal's algorithm is a minimum spanning tree

I want to show that the output spanning tree $S$ of Kruskal's algorithm is a minimum spanning tree, so it is of minimum weight, by contradiction. We suppose that $S$ is not a minimum spanning tree. ...
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1answer
31 views

Have you seen these integer factorization algorithms before?

I have two algorithms for finding two factors, $p$ & $q$, of a number $N$. The algorithms are (hopefully) obviously related. The pseudo-code for them follows: Algorithm 1 ...
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1answer
22 views

Summation to Closed Form conversion

I am struggling to understand basics as it related to forming a closed form expression from a summation. I understand the goal at hand, but do not understand the process for which to follow in order ...
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0answers
39 views

proof that some expected value equal to $\theta (\log n - \log k)$

So here is the problem - Given the following equation: $(c_2\cdot \log n) - (c_1\cdot \log k)\le E(X)\le 1+ (c_1\cdot \log n) - (c_2\cdot \log k)$ When $c_2,c_1\gt0$ and also $c_1\gt c_2$ In ...
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0answers
17 views

how to create sets which has one common value for intersection

There is a algorithm called Maekawa, to send message to a particular request set (http://courses.cs.vt.edu/cs5204/fall99/Supplemental/ProblemSet8.html) I am interested in the ALgorithm to generate ...
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28 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 [A1 .. A2 .... An](random order). We have a comparator C, but it has a probability p to return correct ...
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22 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
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1answer
65 views

The output of Kruskal's algorithm is a spanning tree

I want to show that the output of Kruskal's algorithm is a spanning tree. Let $G$ be a connected, weighted graph and let $S$ be the subgraph of $G$ which is the output of the algorithm. $S$ cannot ...
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1answer
74 views

Is there a fast divisibility check for a fixed divisor?

Is there a fast algorithm to check if $d \mid n$ is true for varying $n$, if divisor $d$ is fixed? Variable $n$ is a $w$-bit binary integer, $d$ is an integer constant.
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1answer
269 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
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0answers
19 views

simple algorithm to go thru all lexicographic permutaions

I'm working thru the euler question 24 (not looking for the answer) and came across this site describing an algorithm to get all the permutaions. Here are the steps they lay out : Steps to ...
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41 views

Is there an optimal algorithm to calculate $2^n - 1$ in $\theta(n^n)$

The sequence $(f_{n})_{n \in \mathbb{N}}$ is defined by $f_{0} := 0, f_{1} := 1$ and $f_{n} := 3f_{n-1}-2f_{n-2}$ for $n \in \mathbb{N}_{0} \setminus \{0,1\}$. Is there an algorithm that takes an $n ...
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4answers
55 views

Two non-negative functions $\,f,g$, such that $\,f \not\in \mathcal O(g)$ and $ g \not\in \mathcal O(\,f)$

Show that there exist two non-negative functions $\,f,g: \mathbb{N} \rightarrow \mathbb{R}$ such that $\,f \not\in \mathcal O(g)$ and $ g \not\in \mathcal O(\,f)$. It would be easy two find two such ...
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23 views

Solving Nonconvex Programming using Genetic Algorithm

Consider such a programming problem, \begin{equation} min \| \Psi(\boldsymbol{x}) \|_p \quad s.t. \quad \boldsymbol{Ax} = \boldsymbol{b} \end{equation} Where $\boldsymbol{x} \in \mathcal{R}^{n}$, ...
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1answer
28 views

Hamilton-like paths in digraph

We are given digraph with two (possibly the same) vertices - let's call them S and F. We are also given some set of vertices W (possibly empty, possibly consisting of all vertices of digraph). We ...
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1answer
60 views

Given the graph below, use Dijkstra’s algorithm to find the shortest path (More details included)

So I've found out a few things and was wondering if someone could verify if I'm doing this correctly. So here is an example I've been given: Here is the solution to that example: Now here is the ...
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1answer
55 views

Need help to understand the solution for NP-completness proof (3CNF <= 0-1 integer-programming problem)

I was trying to solve problem from Cormen page 1100 34.5-2 Given an integer $m * n$ matrix A and an integer $m$-vector $b$, the 0-1 integer- programming problem asks whether there exists an integer ...
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2answers
56 views

Intersection of integer sets

This is probably a trivial question for mathematicians but I am not seeing how to approach the following problem: Imagine two sets defined by: ...
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2answers
28 views

How do I write this equation as an algorithm? [closed]

how do I write "from 3 to 6" or > 2 but < 7 in an algorithm?
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1answer
96 views

Two pawns walking in a complete graph

We have a complete graph $G = \langle V,E\rangle$ with non-negative values on edges. Let $C = \{v_1,v_2,\ldots,v_n\}$ be an ordered collection of $G$'s vertices. At the beginning we have two pawns in ...
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29 views

Baby-step Giant-Step algorithm to calculate value in new base

Using the Baby step–giant step algorithm I am trying to determine $log_{2}(7)$ in base $1$3. Let $p = 7$. Set $n$ to the least integer greater than $\sqrt p$: $n = 3$. So for baby step, I started off ...
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1answer
46 views

Help understanding the complexity of my algorithm (summation)

As an exercise, I wrote an algorithm for summing the all elements in an array that are less than i. Given input array A, it produces output array B, whereby B[i] = sum of all elements in A that are ...
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0answers
22 views

Assignment problem, minization of the Standard Deviation

I have an assignment problem. So typically I need to find the optimal combination between two sets of parameters P, M. I know that the Hungarian Algorithm is often privileged for this kind of problem ...
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0answers
19 views

How to figure out whether PCA can be performed on a data set or not?

I do have idea on the way PCA works but I do not know how to figure out whether a high dimensional data set is suited for PCA compression. I googled for some algorithms but could find any. Are there ...
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15 views

Is there any general algorithm to solve such a 3D cutting problem?

Suppose a cuboid $\mathbb{A}$ has $L$,$M$ and $N$ as its length, width and height respectively, where $L\ge{M}\ge{N}>0$; Now we want to cut $\mathbb{A}$ into smaller cuboids with length $x$, width ...
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1answer
40 views

A result of Erdos: the multiplicative persistence of $n$ is at most $c\ln(\ln n)$

Multiply all the digits of a number $n$ by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence of $n$. ...
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11 views

Spin-off of Scheduling Weighted Interval Problem

I'm trying to solve a problem in which, given a + sign shaped area of land (with no width) and a list of contiguous sections of the land (segments, T-shapes, smaller + shapes, etc), each with an ...
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1answer
40 views

How is de = 1 (mod ϕ(n)) calculated

I am reading RSA algorithm. So, I was writing a question but I saw this question and still couldn't understand it. If $$e\cdot d \equiv 1 \pmod{\varphi(n)},$$ then $$ed=k\cdot \varphi(n)+1, \qquad ...
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1answer
29 views

Proving a recursive algorithm on a set is true

If I have an algorithm that returns the entry of a set with the largest value, how do I prove the algorithm is true mathematically? (I know I could just write tests for it.) I understand how to use ...
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1answer
24 views

How to compare times?

i'm looking for an algorithm to compare 2 different time records. My time format is hour:minute:second:nanosec I tried something like this, but its not working well: if (hour <= anotherHour) { ...
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2answers
40 views

Find Algorithm, given a list of arcs, that maximizes number that fit on a circle

I'm trying to find an optimal algorithm that, given a list of arcs $(x_i, y_i)$, where $x_i$ and $y_i$ are the starting and ending angle measurements of the arc in radians, maximizes the number of ...
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1answer
28 views

Elaboration needed on a section of a published paper about dictionary learning

I will be doing my master thesis on dictionary learning, and I am trying to understand basic concepts of the subject reading the following paper: K-SVD: An Algorithm for Designing Overcomplete ...
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0answers
41 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...