# Questions tagged [algorithms]

Informally, an algorithm is a set of explicit instructions used to solve a problem (e.g. Euclid's algorithm for computing the greatest common divisor of two integers). For more specific questions on algorithms, this tag may be used in conjunction with the approximation-algorithms, algorithmic-randomness and algorithmic-topology tags.

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825 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...
507 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them. Define the set $\mathcal{E}$ of ...
444 views

### What is the geometric intuition behind Wilf-Zeilberger theory?

This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
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### Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
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### Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
302 views

### Fast computation of matrix product $AXA^T$ with fixed $A$?

Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
395 views

### Best known constant for parallel sorting

I recently found myself talking about Szemerédi's mathematics, and briefly discussed his famous sorting network, discovered with Ajtai and Komlós. Apparently their algorithm is not practical because ...
329 views

### Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
155 views

### Are there efficient algorithms to factorise in $\mathbb{N}[X]$?

One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and then to combine some factor to find a factorisation in $\mathbb{N}[X]$. Note that the ...
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### Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
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### Information for discovering an item-colour assignment in a combinatorial game

We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
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### How to find the Eulerian circuit with the minimum accumulative angular distance within a Eulerian graph?

Note: I originally posed this question to Mathematics, but it was recommended that I try here as well. Context For context, this problem is part of my attempt to determine the path of least inertia ...
120 views

### Subgroup membership problem for Noetherian groups

I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group, \begin{...
119 views

### Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...
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### Finding short linear combinations in abelian groups

Let $M$ be a finitely generated abelian group. Assume we are given a presentation of $M$, that is \begin{equation*} M = \frac{\bigoplus_{i=1}^r \mathbf{Z}g_i}{\sum_{j=1}^s \mathbf{Z} r_j} \end{...
241 views

### Parity of number of primes

In https://arxiv.org/abs/1009.3956 is it shown there is a $c>0$ such that $\pi(x)\bmod2$ can be computed in $o(x^{\frac12})$ time (more precisely number of primes $\bmod 2$ for an interval of ...
Take a collection $F$ of homogeneous polynomials in $\mathbb{C}[x_1,\ldots,x_n]$, along with a finite group $G$ of linear operators over $\mathbb{C}^n$ such that for every $f\in F$ and $g\in G$, we ...
I am interested in special graph constructions that have the smallest diameter. We have a path graph $P_n$ ($N$ is even). We add new set of edges $C$ between path nodes such that set $C$ forms a ...