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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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first the trivial facts: Non-degenerate n-dimensional simplices have $n+1$ corners and $\frac{(n+1)n}{2}$ edges. The center of the circum-hypersphere from which all $n+1$ corners are equidistant can ...
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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 ...
joro's user avatar
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According to this lecture on abc https://www.youtube.com/watch?v=zk4U5P61LbM&t=1960s around 18:00 there is a "subtle theorem" that for fixed $k \ge 2$, there are only finitely many ...
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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 $G=(V,A)$ be a directed graph, such that for any two vertices $v,w$, there exists a vertex $u$ such that $(v,u),(u,w)\in A$. Let $s,t\in V$, and let $c:A\rightarrow \mathbb{N}_{\geq 1}$ be arc ...
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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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From our preprint On factoring integers of the form $n=(x^D+1)(y^D+1)$ and $n=(x^D+a_{D-2}x^{D-2}+\cdots+ a_0) (y^D+b_{D-2}y^{D-2}+ \cdots+b_0)$ with $x,y$ of the same size. We got plausible ...
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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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The story of finitary deterministic comparison sorting is well-known: we have the naive $O(n^2)$-time algorithms like selection and insertion sorts; once we try divide-and-conquer recursion we ...
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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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This question was first asked here but got no answer. This paper by R. Garver talks about removing 4 terms from the 9th degree equation. Although everything is easy to understand, there was an ...
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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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Let $a_1(n)$ be A003713, i.e., an integer sequence whose exponential generating function $A_1(x)$ satisfies $$ A_1(x) = \log\left(\frac{1}{1+\log(1-x)}\right). $$ $a_2(n)$ be A141209, i.e., an ...
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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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Let $T(n,k)$ be A111528, i.e., integer coefficients such that $$ T(n,k) = \frac{k}{n} [x^k] \log \left( \sum \limits_{m=0}^{k} m! \binom{n+m-1}{m} x^m \right), \\ T(n,0) = 1, T(0,k) = k!. $$ $R(n,k)$ ...
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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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Given a rational angle $\theta = p/(2^n - 1)$ periodic under doubling, and the corresponding parameter ray $\mathcal{R}^{M}_{\theta}$ of the exterior of the Mandelbrot set, I have two related ...
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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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While experimenting with Newton's method for root finding $x \mapsto x-\frac{f(x)}{f'(x)}$ in positive characteristic, we observed that in the oddest characteristic $2$ there are always short cycles. ...
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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 $$ \operatorname{wt}(2n+1) = \operatorname{wt}(n) + ...
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I am interested in calculating integer and fractional derivatives of a experimental data using discrete Fourier transform. There is a paper Calculating numerical derivatives using Fourier transform: ...
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There are several algorithms for lattice reductions in $n$-dimensions, LLL, etc. Here the lattice in question is in ${\mathbb R}^n$ and the basis vectors $b_1, \ldots, b_n$ are usually assumed to be ...
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Let $H=(V,E)$ be an hypergraph, where $V$ is the vertices set and $E$ is the hyperedge set. A special cycle of $H$ is a sequence $ (v_1, e_1, v_2, e_2, \ldots, v_{k}, e_{k}) $, with $k \geq 3$, where $...
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Let $a$ be an arbitrary non-decreasing integer sequence such that $a(1) = 2$. Here $a(n)$ is $n$-th term of $a$. $b$ be an integer sequence of numbers $k$ such that $a(k+1)>a(k)$. Here $b(n)$ is $...
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Let $T(n,k)$ be A259192, whose ordinary generating function is $$A(x,y) = \frac{1 + xA(xy,y)}{1 - xA(xy,y)}. $$ Start with vector $\nu$ of fixed length $m$ with elements $\nu_i := \delta_{1,i}$, ...
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I'm currently working on a project that partially involves graphs. One of the problems I'm tackling is determining whether two given matrices represent the same connected undirected graph. So given ...
Michael's user avatar
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A common approach to construct fast multiplication algorithms is to make an ansatz for the matrix multiplication tensor of fixed dimension and rank (e.g. $2 \times 2 \times 2$ and rank $7$ if we want ...
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Let $a(n)$ be A032188 (i.e., number of labeled series-reduced mobiles (circular rooted trees) with $n$ leaves (root has degree $0$ or $\geqslant 2$)), whose exponential generating function is $A(x)$ ...
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Let $a(n)$ be A048174 (i.e., number of labeled chains with $n$ edges), whose exponential generating function is $$ \frac{s}{1+s}, $$ where $$ s = \operatorname{SeriesReversion}\left(\log(1+x)-\frac{x^...
user avatar
1 vote
1 answer
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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 $$ \ell(n) = \left\lfloor\log_2 n \right\rfloor .$$ $$ \nu_2(2n+1) = 0, \\ \nu_2(2n) = \nu_2(n) + 1. $$ $a(n)$ be A003159 (i.e., numbers whose binary representation ends in an even number of ...
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Let $T(n,k)$ be an integer coefficients such that $$ T(n,k) = \begin{cases} 1 & \text{if } k = 0 \vee n = k \\ \displaystyle{ \sum\limits_{i=k-1}^{n-1} T(n-1,i) T(i,k-1) } & \text{otherwise} \...
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Suppose $A$ is an $n \times n$ dimensional Hermitian matrix with $\|A\| \le 1$. I now consider the QR algorithm. I set $A_0 = A$ and at the $k$th step compute the QR decomposition $A_k = Q_k R_k$ and ...
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I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
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Define a $d_0\times d_1\times\cdots\times d_{k-1}$-grid to be a surjective function $G$ with domain $\prod_{i=0}^{k-1}\{0,1,\dots,d_i-1\}$. A $d_0\times d_1\times\cdots\times d_{k-1}$-grid can also be ...
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Suppose we have a set $N$ of $n$ players and a set $M$ of $m$ items. We are given a matrix $P_{n \times m}$ whose element $p_{ij} \geq 0$ $(i \in N, m \in M)$ denotes the valuation of good $j$ by ...
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Given a finite poset $(X, \prec)$ of size $n = |X|$, how efficiently can we compute a constant-factor approximation to the maximum antichain? It is well-known that the size of the largest antichain ...
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I am trying to calculate the time complexity of the Floyd-Warshall algorithm with the following modification: Instead of adding a single intermediate node each time, I will be adding $K$ nodes at once....
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Let $T(n,k)$ be A253829 whose ordinary generating function is $$ A(x,z) = \prod\limits_{n=1}^{\infty} \frac{1-z}{1-z-xz^n}. $$ $\nu$ be a vector of fixed length $m$ with elements $\nu_i = \delta_{1,i}...
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I've been looking at a specific optimization algorithm in the nonconvex setting, and I'm trying to analyze its convergence rate. Here's the quick setup: Suppose we have a nonconvex, continuously ...
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This came when examining edge labeled graphs and is based on limited experimental evidence so counterexamples are likely. Let $T(G)$ denote the treewidth of a graph. Transformation (1) Set $G'=G$ and ...
joro's user avatar
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Let $a(n)$ be A103239. Here $$ 1 = \sum\limits_{n=0}^{\infty} a(n) \frac{x^n}{(1-x)^n} \prod\limits_{j=0}^{n} (1 - (j+2)x). $$ Start with $A$ and vector $\nu$ of fixed length $m$ with elements $\nu_i =...
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Let $a(n)$ be an integer sequence with ordinary generating function $A(x)$ such that $$ A(x) = \prod\limits_{k=1}^{\infty} \frac{1}{1+\frac{(-x)^k}{1-x}}. $$ Start with vector $\nu$ of fixed length $m$...
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1 answer
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Basic setups: A Turing machine M operates on a doubly infinite tape. The tape is divided into cells. Each cell can hold either a 0 or a 1 (meaning, we will consider only TM’s with two symbols). The ...
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Fix distinct primes $q_1,\dots,q_t\in[2^{m-1},2^m]$ and integers $r_i\in[0,q_i-1]$ at every $i\in\{1,\dots,t\}$. Is there a way to exactly count the number of primes $a\equiv r_i\bmod q_i$ where $a\...
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I have a polynomial of degree around $2^{35}$ over a finite field $\mathbb{F}_p$, where $p$ is a 64-bit prime number. What is the most efficient algorithm for finding a root of such a polynomial? (...
user's user avatar
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Question: what is known about calculating from the real-valued coefficients of polynomial $P(x)$ in monomial form, i.e. $P(x)=\sum\limits_{i=0}^n c_ix^i$ the polynomial $Q(x)=\sum\limits_{i=0}^nd_i(...
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Let $F(n)$ be A000045 (i.e., Fibonacci numbers). Here $$ 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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10 votes
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When discussing the correctness, accuracy, condition, stability etc. of numerical algorithms, the arguments are usually performed in $\mathbb{R}$. Only when considering stability in the presence of ...
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