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Questions tagged [algorithms]

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

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Algorithm to find at least twice greater nearest right element for every element in array

Given array of positive real numbers $a[1, \dots, n]$, we need to provide an algorithm to find at least twice greater nearest right element for every element in array. My first thought is to make an ...
5
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1answer
45 views

A polynomial algorithm to determine whether a finite group is nilpotent

Does there exist a polynomial (in respect to the order of the group) algorithm that given a Cayley table of a finite group determines, whether a group is nilpotent or not? There do exist polynomial ...
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0answers
32 views

Primitive Factors of Mersenne numbers via Binary Programming

The problem I am considering is a general binary-programming one, but permit me to introduce it here with a specific example: Given integer $n = 295$, find the first Mersenne number, $M_{k} = 2^{k}...
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BPP-P, GA or something else

I would like to preface this question by apologising in advance if it is stupid/completely wrong and thank you in advance for being patient with someone new to this field. I am looking for the best ...
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1answer
20 views

Play Cards Game Tournament Algorithm

I am currently trying to find algorithm to minimize the total time of a tournament. The game requires $2$ teams of $2$ players in each team (total $4$ players). Then, the perfect number of ...
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1answer
23 views

Complexity of solving a linear equation system over $k[x]$

Let $k$ be a field and let $A \in k[x]^{m \times n}$ be a polynomial matrix whose entry with highest degree has degree $d$. Let $b \in k[x]^m$. What is known about the complexity of computing a ...
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34 views

Dinic's algorithm

Given $ N = (G = (V , E),s, t, c)$ a flow network, we run the Dinic's algorithm (https://en.wikipedia.org/wiki/Dinic%27s_algorithm) on this Network. Consider some iteration $i$ which is not the last ...
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1answer
53 views

algorithms - When not to assume $n$ is a power of 2, while solving recurrences?

In Udi Manber - Introduction to Algorithms. A Creative Approach, exercise 3.29, p.59 Although in general it is sufficient to evaluate recurrence relations only for powers of 2, that is not always ...
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2answers
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Recurrence - Master theorem [on hold]

I need to solve $T(n) = 4T(\frac{n}{3}) + n\log(n)$ To apply the master theorem to the function, I need to find $a$, $b$ and $d$. Then $a$ is $4$, $b$ is $3$ but what is $d$? I have never encountered ...
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0answers
66 views

Efficient method to check whether the nearest prime has distance $d$ or more?

Suppose, a prime $\ p\ $ is given. How can I check efficiently whether the distance to the nearest prime is $\ d\ $ or more , if $\ d\ $ is given ? My approach is to start with $\ c=2\ $ and as ...
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0answers
16 views

Tough test polynomials for (finite precision) complex root finding methods, especially Aberth's method

Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing ...
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2answers
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Solving recurrences with Master theorem

Normally, the way I am solving those problems is the following: $3T(\frac{n}{2})+ n^4$ $a=3 ; b = 2 ; d=4$ then I am doing $\log(b(a))$ which is $log(2(3))= 0.6$ Since $0.6 < d$ I can apply the ...
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0answers
18 views

asymptotic growth of functions

I ordered a list of function based on asymptotic growth but I am not 100% not. Faster to slower: n^0.001 (√n ln n) 2^(ln^2n) 2^(2^ln n) (ln ln n^2) (ln n)! n!
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Path between nodes in a colored oriented tree with given weight sum

Consider an oriented tree where each node is colored either black, white, or both. In addition, each (oriented) edge has a given weight. I am trying to see whether there exists a pair $(u, v)$ of ...
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1answer
21 views

A standard way to reduce a k-SAT to 0-1 Integer Linear Programming

I was searching for a standard (a published paper) for which it reduces a k-SAT to a 0-1 ILP (Integer Linear Programming), but couldn't find any :( I know how to reduce a SAT problem to an ILP ...
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0answers
35 views

Derive a bound for a tree with node having k left branches

We are given a binary tree of maximum level n and where each node can have a maximum $k$ left directed edge. $n$ is always greater than or equal to $k$. I want to know a bound on the number of nodes ...
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0answers
36 views

Choosing vectors for projection

I have a vector $v = a_1\mu_1 + a_2\mu_2 + ... + a_n\mu_n$ where $\mu_i$ are given linearly independent but not orthogonal vectors. I need to choose $k$ vectors from the original set such that when ...
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Finding optimum time for change of equipment based on shift pattern available resources

I am trying to sort out a relatively simple problem, where I believe an algorithm or solving technique may already exist (Hungarian Algorithm?). I'd like to solve it using an methodology rather than ...
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0answers
57 views

Consider a binary operation $\circ:\{1,2,\ldots,n\}\times\{1,2,\ldots, n\} \rightarrow \{1,2,\ldots, n\}$.

Consider a binary operation $\circ:\{1,2,\ldots,n\}\times\{1,2,\ldots, n\} \rightarrow \{1,2,\ldots, n\}$. Сall the degree of associativity of this operation the number of triples $i, j, k$ such ,that ...
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52 views

Test if a function is continuous or has at least one discontinuous vertical asymptote between an interval

Imagine evaluating a function with little intervals incrementally across a graph and testing by using the end points of the each interval (and maybe a midpoint), whether the function is continuous for ...
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0answers
27 views

Find polygon that traverses all given points with minimal circumference but has a single surface

I know about Convex Hull, but convex hull is convex, which isn't what I want. I want a polygon that will always try to minimize it's perimeter however will not have multiple surfaces: For example, I ...
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1answer
31 views

How can I properly force this “object” to follow the indicated path

I'm working with drones and I want them to follow a path. This is the current behavior (black is the path, red is the drone): As you can see, it goes to the destination point (C), but it doesn't ...
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2answers
53 views

(n & (n -1)) == 0 [closed]

I am going through this interview study book for algorithms and it states that: " So, we have our answer: ( (n & (n -1)) == 0) checks if n is a power of 2 (or if n is 0). " How is this possible, ...
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0answers
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Minimal one-step distance set of vertices to all vertices

I want to find minimal set of vertices such that every vertex in this graph either in this set or connected with some vertex from this set with one edge. Is there standard name for this algorithm? It ...
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1answer
25 views

Cut In a Flow Network

Given $ N = (G = (V , E),s, t, c)$ a flow network (assume that the capacity $c$ is always positive) and $e = (u,v) \in E$. I would like to develop an algorithm that tell if there exist a min-cut (cut ...
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1answer
9 views

Algorithm Problem for arrival and departure time of a dataset

Algorith for arrival and departure time I have got a piece of algorithms which consists of several equations to determine the arrival and departure times of a water vessel to a dock from AIS data. I ...
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0answers
33 views

Finding small solutions to modular congruences

I was wondering what computational/algorithmic techniques can be used to solve a modular congruence when we are looking for a pair of small values. The specific problem is like this (the numbers are ...
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1answer
32 views

time the pseudo random generator gonna start repeating itself

as you know the general formula for pseudo random generator is this U(n)=a*U(n−1)+b [mod z] where we have control of U(n-1)...
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0answers
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Which uniformed search algorithm is the best here?

So I have an exercise like this where I have to choose the best strategy. I wonder here which one is the best: A robotic engineer wants to add movement planning ability to a small mobile robot. ...
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2answers
45 views

Biased binary search complexity

We know Binary search on a set of n element array performs O(log(n)). We have this recursive equation through which the search space is reduced by half in each iteration, after a single comparison. <...
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Compute the shortest path in a directed acyclic graph

Problem: Let $D = (V,A)$ be a directed acyclic graph, i.e., there exists no directed cycle in D, and let $w : A → R$ be arc weights. Assume that you are given a topological sort of the vertices. Show ...
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27 views

Algorithm expressing a fraction as a sum of unit fractions

I am looking for an algorithm to express some fraction $\frac{a}{b}$, with $a,b \in \mathbb{Z}$, as a sum of unit fractions, like: $$\frac{a}{b} = \frac{1}{w_1} \pm \frac{1}{w_2} \pm \frac{1}{w_3} \...
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1answer
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Find all rational solutions of 1) $3x^2 +8x^2 -15x+4=0$ question 2)$x^4 -6x^3+22x-30x+13=0$ [closed]

Finding all rational solutions of $3x^2 +8x^2 -15x+4=0$ and $x^4 -6x^3+22x-30x+13=0$
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2answers
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Time complexity for an algorithm

How do you realize that $$(3log^2(n) + 55log(n^{10})+8log(n))*log(n) \neq \Omega(log^{10}(n))$$ ,where $log^x(n)$ means $(log(n))^x$ I know that by definition, if $f(n) = \Omega(g(n))$ ...
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1answer
34 views

are $\sum_{i=0}^{n}i$ and $\sum_{i=n}^{0}i$ equivalent?

So here's the ugly history of how I came to ask this question. I was following this proof: and got stuck at this step: $$\sum_{j=0}^{(\log_2n) - 1}\frac{1}{(\log_2n) - j} = \sum_{l = 1}^{\log_2n}\...
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0answers
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LLL for quadratic forms

It is quite easy to use the LLL algorithm to find approximate solutions to linear/multilinear forms, and I am able to do that. However, I am trying to understand how the LLL algorithm is used to ...
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0answers
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Measure effectiveness of algorithm (plot included for clarification)

I have written a short algorithm that computes the "comovement" of a time series. My problem is what method to use to measure how accurate this algorithm is. Ideally it should only have negative lines ...
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1answer
17 views

Calculate generators of an intersection of Ideals

$$I = (x_1^2-x_1,x_2^2-x_2,...,x_n^2-x_n,t-\sum_{i=1}^n 2^{i-1}*x_i)$$ Ideal in $\mathbb Q[x_1, ..., x_n,t]$. How can I calculate the generators of $J = I \cap \mathbb Q[t]$ by hand? I tried it with ...
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0answers
15 views

Reducing knapsack problem to a inversed knapsack problem

1)Suppose we have a common 0-1 knapsack problem. Given a set of n items numbered from 1 up to n, each with a weight w_i and a value v_i, along with a maximum weight capacity W. Here we need to select ...
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How many swaps in a set of size n will ensure that the set is shuffled reasonably well?

I'm implementing my own version of a shuffle method for shuffling a set of objects in a list. My implementation generates two (pseudo)random numbers and swaps the elements at these two indexes. ...
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0answers
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Discret optimization problem

the problem setting is as follows: We are given a set of words (for simplicity we can assume all of the same length, i.e. n-tuples) $W = \{ w_1, w_2, ..., w_N \}$ where $w_i = (a_{i1}, a_{i2}, ... a_{...
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Can a network flow have a vertex with no s-t path? [closed]

I was wondering if the second picture was a valid example of a network flow? I don't think it is since I haven't actually seen an example of such a graph but I could be wrong?
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Domination problem is NPC

I call the problem as DOM. Given a graph $G$ and an integer $d$, Decision problem DOM - "Does there exists a dominating set of size less than or equal to 'd' in G? " DOM is NP Complete problem. But, ...
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3answers
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How come the time complexity of Binary Search is log n

I am watching this professor's video on Binary Search but when he reached here, I am a bit lost. How come he came up the time coomplexity is log in just by breaking off binary tree and knowing height ...
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How to show time complexity comparation of Fleury's and Hierholzer's algorithm?

I know that time complexity of Fleury's algorithm equals to $O(e^2)$ and Hierholzer's equals to $O(e)$. I need to show that Fleury's algorithm is less efficient on some example or by giving a proof of ...
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0answers
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The number of colours in greedy colouring is $\leq \frac{1}{2} + \sqrt{2E(G)+\frac{1}{4}} $

I am stuck with the following exercise: Let $\sigma$ be an ordering of the vertices of a given graph $G$ and let $\chi(G,\sigma)$ be the number of colours the greedy colouring algorithm uses. Show ...
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0answers
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Gaussian Elimination Roundoff Error Confusion

I have a couple questions about Gaussian Elimination Regular Gaussian Elimination (No pivoting, No making pivots 1, No Row Swapping) Gaussian Elimination with pivoting (Which just means row ...
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2answers
52 views

Easy ways of finding small prime divisors

Are there any "relatively simple" ways of seeing whether a number is divisible by a small prime not 3. For example, one simply sums the digits of n modulo 3 and if they sum to zero, then n is ...
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0answers
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Looking for ways to transform time-series data recorded from object movement into equation describing the movement direction of the object

Looking for some time-series data transformation advice! I want to know what's the best way to transform data of 9-tuples time series data of IMU (Inertia Measurement Unit) sensor, recorded from a ...
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0answers
38 views

Recurrence relationship for crawling a directory

I am trying to write a recurrence relationship for problems that can be solved using recurrence. As an example recurrent for finding the 3^4 (which is 3*3*3*3) can be written as: ...