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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12
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130 views

Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
9
votes
0answers
175 views

Algorithm for obtaining the surface of a mirror

My colleague and I have been trying to implement an algorithm described in the paper "Recovering local shape of a mirror surface from reflection of a regular grid", primary author of which being ...
9
votes
0answers
596 views

A constrained topological sort?

Suppose that one has a directed, acyclic graph G, and each vertex $v$ contains a (positive) value $a_v$. Additionally, let $r$ be a constant. For my purposes, $r>1$, but this might not matter. ...
8
votes
0answers
100 views

Algorithm to compute fastest method of collecting $k$ re-spawning items which spawn at $n$ specified points

Let $V = v_1, \dots, v_n$ be the locations the items can spawn at, and let $U = u_1, \dots, u_k$ be the current positions of the items. We will assume a new items spawns instantly every time we ...
7
votes
0answers
366 views

Problem with an algorithm to $3$-colour the edges of cubic graphs

I'm currently trying to implement an algorithm to $3$-colour the edges of cubic graphs. (I want to do this with Matlab's Symbolic toolbox). After restricting to planar graphs to ensure the existence ...
7
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0answers
193 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
7
votes
0answers
194 views

What's the most efficient algorithm for Divisibility?

What is the most efficient (in time complexity) algorithm known nowadays for the Divisibity Decision Problem: given two integers, say $a$ and $b$, does $a$ divide $b$? Let it be clear that what I ask ...
7
votes
0answers
188 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
7
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0answers
299 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
6
votes
0answers
47 views

Is there a numeral system that makes both addition and multiplication easy?

Decimal positional notation, the system for writing numbers we all use every single day, makes addition very easy by transforming it from a computation to a repeated operation on individual digits ...
6
votes
0answers
86 views

Quickest way to solve a matrix one step at a time.

I have a $14\times14$ matrix with a possibility of six states in each position The matrix is random each time. An example matrix would be: $$ \begin{pmatrix} ...
6
votes
0answers
551 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
6
votes
0answers
463 views

Hardness of finding eigenvalues over finite fields

How hard is it (computationally) to find eigenvalues/eigenvectors of matrices over finite fields? Suppose the field has size exponential in the input. (Does the QR algorithm still converge?) How ...
5
votes
0answers
29 views

Computing N cofactors

I have an $N\times N$ matrix of small size (say $N=20$). Is there a way to evaluate the $N$ cofactors of the elements of the first row, faster than in the obvious way as $N$ independent determinant ...
5
votes
0answers
48 views

A summation involving the ceiling function

I'm trying to find a better method of calculating the sum $$\sum_{k=1}^N\lceil ak\rceil^2$$ where $a$ is an irrational number. So far, my only idea is to somehow use a best rational approximation. ...
5
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0answers
96 views

Traveling salesman problem: can a terrible strategy beat a good one?

Until yesterday, I was under the naive impression that constructing a weighted graph where the nearest-neighbour algorithm gives the worst possible route, would have the property that any other ...
5
votes
0answers
48 views

Decoding of Gabidulin code

Consider the space of matrices in $\mathbb{F}_q^{n \times m}$ where $\mathbb{F}_q$ is the finite field with $q$ elements. We can define a metric on this space, given by $d(A,B) := rank(A-B)$, called ...
5
votes
0answers
218 views

Generating a stochastic matrix with a given second dominant eigenvalue

I need a procedure (iterative or otherwise) that, given a positive integer $N$ and a (possibly complex) number $\lambda$ such that $0 < \vert \lambda \vert < 1$, will be able to generate an $N ...
5
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0answers
108 views

Nice puzzle: Creating a binary word using weights and scales

You are given $N$ weights where for each $i \in \{1,2,...,n\}$, the $i$-th weight weighs $i$ pounds. You are given an $N$-binary-word that's formed by $L$'s and $R$'s and scales. You need to provide ...
5
votes
0answers
137 views

$k$-covering of the set of all possible n-length words

Give an alphabet $\mathcal{A}=\{a_1,a_2,\ldots,a_m\}$, and let $L_n$ is the set of all possible $n$-length words in form $[a_{i_1}a_{i_2}a_{i_3}\ldots a_{i_n}],\ a_{i_j}\in \mathcal{A}$. We call a ...
4
votes
0answers
19 views

Simple criteria to know if the p-nary notation of an integer can generate a tree by preorder traversing?

I am treating with a preorder tree traversal structure(which means sequences where the children of each tree node are listed behind it) now for some other problems and the structure is like: ...
4
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0answers
45 views

Algorithm for finding “fact families”

My friend's 3rd grader encountered the following question regarding "fact families" on her math homework: I was in 3rd grade sometime in the 1980s, so I don't believe I ever encountered this term ...
4
votes
0answers
73 views

Is there a way to figure out the minimum number of participants or maximum number of rounds in my tournament style?

I just finished hosting a Euchre tournament at work that was meant to get people to meet other people in the company. This is the third time I've hosted this type of tournament. The first two times, ...
4
votes
0answers
30 views

Minimum number of vertex moves to un-intersect a polygon with itself

In my game, I have n points that form a self-intersecting polygon. The points can be moved by dragging them. How can I form a non-intersecting polygon this way, in ...
4
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0answers
103 views

Binary Palindrome

Let an integer n be given. Write the integers from 1 to n in binary notation successively from left to right. In the resulting string consisting of zeros and ones, choose a palindrome substring of ...
4
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0answers
106 views

Computational hard math problem

Given a square filled randomly with the numbers $1$ to $N$, for instance $$\begin{array}{cccc} 16 &12 & 9 & 1\\ 11 & 3 & 4 & 7\\ 2& 8 & 5&14\\ 6& 10& ...
4
votes
0answers
56 views

Selecting k vectors with maximum spread out of a set of n vectors

Given a set $\mathcal{V}$ of $n$ vectors, find a subset $\mathcal{V}_k = \mathcal{V} - \mathcal{V}_{n-k}$ containing $k$ maximally spread vectors. Intuitively, these $k$ vectors should be spread as ...
4
votes
0answers
377 views

How to reverse this bitwise AND-XOR encoding algorithm?

I have been given an "encoding" algorithm that does bitwise XOR and bitwise AND. Originally it's a C code that operates on integers with bit-shifts, but I have translated it into a simpler pseudocode ...
4
votes
0answers
194 views

Test for equivalence of algebraic expressions

We are looking for the most efficient (most recent, or best) techniques to check if two algebraic expressions (elementary, Calculus-type functions) are equivalent (or if an expression is equivalent to ...
4
votes
0answers
113 views

What is the name of this algorithm for calculating the lowest common multiple?

I have implemented a quite slow algorithm to calculate the lowest common multiple of 20 different integer values. It is described here with an example : ...
4
votes
0answers
158 views

Use of graph theory to determine tensor contraction ordering

I am considering using a computer program to execute tensor contractions like the following: $\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L} $ ...
4
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0answers
331 views

algorithm for solving diagonal quadratic equations over real or complex numbers

I found the following statement in the paper http://www.math.uni-bonn.de/~saxena/papers/cubic-forms.pdf (page 22, in the middle): For $\mathbb F\in\{\mathbb R, \mathbb C\}$ and $b, a_i\in\mathbb ...
4
votes
0answers
372 views

Random binary invertible matrix

For implementation of McEliece cryptosystem, I'm trying to generate a random binary invertible matrix and its inverse. Because this is usually the most time-consuming part of generating a McEliece ...
4
votes
0answers
168 views

Convex hull of balls

The convex hull is defined as the smallest convex set containing a set of points. Now we want to generalize it to a set of balls. If these balls have the same radius, then it can be shown that a ball ...
4
votes
0answers
416 views

When do floors and ceilings matter while solving recurrences?

I came across places where floors and ceilings are neglected while solving recurrences. Example from CLRS (chapter 4, pg.83) where floor is neglected: Here (pg.2, exercise 4.1–1) is an example ...
4
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0answers
504 views

Graph theory classic problem

Suppose there is a city,the city has N buildings. In fact, any building in this city is either A HOUSE or A RESTAURANT. The N buildings are conveniently numerated by integer numbers from 1 to N. To ...
4
votes
0answers
693 views

Project Euler Problem 338

I'm stuck on Project Euler problem 338. This is a cross post from StackOverflow where I initially posted, however, it was suggested that I post it here too since the problem mostly relies on math. The ...
4
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0answers
310 views

Bellman-Ford algorithm with changes

I got this question and I will be happy for a clue. Here is a similar algorithm to the Bellman-Ford algorithm: ...
4
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0answers
269 views

Optimal sorting algorithm with modified cost

First, this was one of the four problems we had to solve in a project last year and I couldn’t find a suitable algorithm so we handle in a brute force solution. Problem: The numbers are in a list ...
3
votes
0answers
19 views

Finding lower bounds for solution of linear Diophantine equation in two varuables

I am currently encountering the following arithmetical problem : given four nonzero integers $A,B,C,D$, let $\Omega=\bigg\lbrace (x,y)\in{\mathbb N_{\geq 0}}^2 \ \bigg| \ \frac{Ax+By}{Cx+Dy} \in ...
3
votes
0answers
43 views

Algorithm for Converting Rational Into Surreal Number

I'm writing a library in Haskell that represents the class of surreal numbers, which like many things can be read about here. I've run into a problem in converting between other classes of numbers ...
3
votes
0answers
49 views

$n$ solutions Sudoku

Is there an algorithm which can calculate a $9 \times 9$ Sudoku with non-trivial $n$ possible solutions? So if you play it you can play it for example 4 times? So if you play it there's a choice in ...
3
votes
0answers
36 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
3
votes
0answers
93 views

a problem about finding an algorithm for a spanning tree in a 3-regular graph

"Consider the connected 3-regular graph G. Find an algorithm that produces a subgraph S of G which is a spanning tree and if you remove S from G then G is divided into some components that each of ...
3
votes
0answers
40 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
3
votes
0answers
49 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
3
votes
0answers
25 views

Looking for an algo to “sorta” diagonalize a similarity matrix.

I've got a big fat similarity matrix. The rows and columns represent people, and the values represent some positive measure of their closeness (0 meaning no connection at all). The n-th row and n-th ...
3
votes
0answers
142 views

Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
3
votes
0answers
47 views

Speed dating/networking challenge

I am trying to organise an event with 54 participants. I want them to participate in 9 different activities at stations around a hall. Obviously this will require 9 sessions to allow the participants ...
3
votes
0answers
15 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...