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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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Motivation. In my younger son's class, everyone has to give a (small) Christmas present to one other student. Let $n\in\mathbb{N}$ be the number of students in the class. If you pick a permutation $\...
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Factorization of integers of special forms are of both theoretical interest and cryptographic implications. Experimentally we found a seemingly "large" set of integers for which a divisor ...
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Let $x,y,X,Y,D>1$ be positive integers. Let $X^D+Y^D=n$ and assume $Y>X^{D/(D-1)}$. Conjecture 1 $Y=\lfloor n^{1/D}\rfloor$ and $X=(n-Y^D)^{1/D}$ Let $Y^D-X^D=n$ and assume $Y>X^{D/(D-1)}$. ...
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Let $p,C$ be positive integers and assume $0 < C< \sqrt{p}$ Let $n=p (p+C) $ and assume $n$ is odd. Conjecture 1 Given $n$ we can find $p,p+C$ with complexity computing two integer square roots ...
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Let us consider two monic polynomials $f(X), g(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]$. Now, we call $h(X)$ is a divisor of $f(X)$, if there exists a $l(X) \in \dfrac{\mathbb{Z}}{p^k\mathbb{Z}}[X]...
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We are provided with a set of $n$ targets. Each target is characterized by a utility value. We know the distribution of the utility value for each target, but do not know its current value. Therefore, ...
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Let the following data be given. Two positive integers $m$ and $n$. A family of sets $B_a \subseteq \{1, \dots, n\}$ (for $a \in \{1, \dots, m\}$). The task is to count the number $N$ of injective ...
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I have tried to implement Ramanujan's algorithm for Solvability of a system of polynomial equations but got stuck in the final step of calculating the partial fraction decomposition from which the ...
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The origin of the following question is irrelevant and will hence be skipped. Hopefully, it will be interesting without context. Consider the following algorithm. We will use $p_i$ to denote the $i$-...
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Question: what is the fastest algorithm for optimally triangulating a point set $P$ that is given as the union of two sorted sequences $A\cup B\ :=\ ([0,0],[x_{a_1},0],\,\dots,\,[x_{a_m},0],[1,0])\,\...
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Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor. $$ $\operatorname{wt}(n)$ be A000120, i.e., number of ones in the binary expansion of $n$. Here $$ \operatorname{wt}(2n+1) = \operatorname{wt}(n) + ...
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There are several algorithms for lattice reductions in $n$-dimensions, LLL, etc. Here the lattice in question is in ${\mathbb R}^n$ and the basis vectors $b_1, \ldots, b_n$ are usually assumed to be ...
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Let $a$ be an arbitrary non-decreasing integer sequence such that $a(1) = 2$. Here $a(n)$ is $n$-th term of $a$. $b$ be an integer sequence of numbers $k$ such that $a(k+1)>a(k)$. Here $b(n)$ is $...
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I'm currently working on a project that partially involves graphs. One of the problems I'm tackling is determining whether two given matrices represent the same connected undirected graph. So given ...
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Let $a(n)$ be A032188 (i.e., number of labeled series-reduced mobiles (circular rooted trees) with $n$ leaves (root has degree $0$ or $\geqslant 2$)), whose exponential generating function is $A(x)$ ...
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A graph $G$ is said to be a tree if it is a connected (i.e. every two distinct vertices of $G$ can be connected by a path of vertices of $G$), acyclic (i.e., $G$ has no cycles) and simple (i.e., $G$ ...
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Let $T(n,k)$ be an integer coefficients such that $$ T(n,k) = \begin{cases} 1 & \text{if } k = 0 \vee n = k \\ \displaystyle{ \sum\limits_{i=k-1}^{n-1} T(n-1,i) T(i,k-1) } & \text{otherwise} \...
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I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
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Suppose we have a set $N$ of $n$ players and a set $M$ of $m$ items. We are given a matrix $P_{n \times m}$ whose element $p_{ij} \geq 0$ $(i \in N, m \in M)$ denotes the valuation of good $j$ by ...
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I am trying to calculate the time complexity of the Floyd-Warshall algorithm with the following modification: Instead of adding a single intermediate node each time, I will be adding $K$ nodes at once....
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I've been looking at a specific optimization algorithm in the nonconvex setting, and I'm trying to analyze its convergence rate. Here's the quick setup: Suppose we have a nonconvex, continuously ...
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Let $a(n)$ be A103239. Here $$ 1 = \sum\limits_{n=0}^{\infty} a(n) \frac{x^n}{(1-x)^n} \prod\limits_{j=0}^{n} (1 - (j+2)x). $$ Start with $A$ and vector $\nu$ of fixed length $m$ with elements $\nu_i =...
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Basic setups: A Turing machine M operates on a doubly infinite tape. The tape is divided into cells. Each cell can hold either a 0 or a 1 (meaning, we will consider only TM’s with two symbols). The ...
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I have a polynomial of degree around $2^{35}$ over a finite field $\mathbb{F}_p$, where $p$ is a 64-bit prime number. What is the most efficient algorithm for finding a root of such a polynomial? (...
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Let $F(n)$ be A000045 (i.e., Fibonacci numbers). Here $$ F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1. $$ Let $a(n)$ be A268407 (i.e., number of North-East lattice paths that do not bounce off the ...
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