The algorithms tag has no wiki summary.

**2**

votes

**1**answer

215 views

### Complexity of establishing finite groups (non)-isomorphism ?

Question Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N ...

**8**

votes

**3**answers

1k views

### Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ?
(Main case - complex numbers, comments on other cases are also welcome. "Given" ...

**2**

votes

**2**answers

238 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**0**

votes

**1**answer

93 views

### Provided a list of sets, $L$, computing an array where each entry $q_i \in Q$ is the family of sets in $L$ that have intersection $k$ with $l_i \in L$

I have a set of $(l_1, ..., l_N) \in L$ smaller sets, each with $(r_1, ..., r_M) \in R$ integer elements. I would like create an ordered array of $(q_1, ..., q_N) \in Q$ sets s.t.:
(1) Each $q_i \in ...

**9**

votes

**1**answer

273 views

### Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C)

Let $\Gamma = \pi_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.
Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix ...

**1**

vote

**1**answer

244 views

### Computing the ratio of two large integers modulo m

$P(n)$ and $D(n)$ are two large integers.
Suppose $R(n) = \frac{P(n)}{D(n)}$ is an integer.
I want to compute $R(n)\bmod m$.
$P(n)$ and $D(n)$ are too large to be computed but $P(n)\bmod m$ and ...

**0**

votes

**1**answer

457 views

### Finding the lowest cost set of disjoint paths using all nodes in a directed graph?

I have a directed graph with edges connecting nodes representing costs.
I wish to find the set of paths which
-go from node 'start' to node 'end'
-are node-disjoint (except for the start and end ...

**1**

vote

**5**answers

1k views

### The Inverse of the Euler Totient Function

How can we calculate the cardinality of the inverse of Totient function of any positive integer n ?
I tried going through this paper, but I couldn't understand the procedure.
Thanks

**1**

vote

**0**answers

122 views

### Efficient algorithm for computing the integral closure of a computable domain

what is known? even talking about efficiency relatively to the complexity of the computation of the domain itself?

**6**

votes

**2**answers

408 views

### Algorithm for reducing words in a Coxeter group

Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is ...

**1**

vote

**0**answers

89 views

### Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?

Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Equally interesting would be to learn about such problems with a ...

**-5**

votes

**2**answers

620 views

### why do we need algorithms, and why is non-convex optimization difficult? [closed]

A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...

**9**

votes

**2**answers

611 views

### How to compute the Picard rank of a K3 surface?

I'm curious about the following question:
Given a K3 surface, how does one proceed to compute its rank?
Of course the answer may depend on the form of the input, i.e. how the K3 is "given". So
...

**11**

votes

**1**answer

1k views

### Conceptual explanation of Strassen's trick for matrix multiplication

Algorithms for fast multiplication of polynomials and integers have well-known conceptual explanations. A good survey paper is Daniel J. Bernstein's Fast Multidigit Multiplication for Mathematicians.
...

**2**

votes

**0**answers

162 views

### A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete.
I provide two simple definitions and one example at first.
Definition 1.
Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...

**6**

votes

**1**answer

1k views

### Origin of square-and-multiply algorithm

I'm teaching an introductory course in cryptography and explained the square-and-multiply algorithm to the class.
http://en.wikipedia.org/wiki/Square-and-multiply_algorithm
Someone asked who ...

**0**

votes

**1**answer

239 views

### Counterexamples for this algorithm for recognizing lexicographic product of graphs?

Found a possible reduction from recognizing lexicographic product of graphs to 2SAT
(since 2SAT is polynomial, the algorithm is polynomial).
Can't prove completeness of the algorithm and since it is ...

**10**

votes

**3**answers

428 views

### Algorithms in Invariant Theory

Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...

**2**

votes

**3**answers

1k views

### Algorithm for detecting prime powers

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote:
"This scheme will thus work as ...

**5**

votes

**0**answers

321 views

### Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?

**5**

votes

**0**answers

192 views

### Rigid-body matching algorithm and clustering algorithm with groups of lines in 3D

I've been struggling with this problem for weeks, and couldn't find an appropriate algorithm to solve it. Could you guys please give me some advices or suggestions in addressing this question. Or if ...

**1**

vote

**0**answers

135 views

### Randomized alternative to Buchberger's algorithm

Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...

**0**

votes

**1**answer

234 views

### Algorithm for vector space

I have $n$ vectors $e_1 \in (\mathbb Z/2 \mathbb Z)^m,\dots,e_n \in (\mathbb Z/2 \mathbb Z)^m $
and a vector $ v \in (\mathbb Z/2 \mathbb Z)^m $
I need to find the better algorithm which answers ...

**2**

votes

**2**answers

990 views

### Generation of All Path in a Directed Acyclic Graph

I am working on a very large dataset of a single DAG whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My ...

**1**

vote

**1**answer

185 views

### On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that
$$
A_0 = A_1 = 0;\quad A_n = ...

**1**

vote

**1**answer

121 views

### How to deconstruct a sum of intersecting upsets

A set system $\mathcal{U}\subset P([n])$ is
an upset if $B\supset A \in \mathcal{U}$ implies $B\in \mathcal{U}$,
intersecting if $A,B\in\mathcal{U}$ implies $A\cap B \ne \emptyset$.
Note that a ...

**3**

votes

**2**answers

312 views

### pseudo-random algorithm allowing O(1) computation of Nth element

It is obvious that using seed value one can easily compute next value of some (deterministic) pseudo-random algorithm - so Nth element can be computed in O(N).
But is there such PRNG that allow to ...

**0**

votes

**1**answer

159 views

### Designing a Hash function that results in minimum size and no collisions

Dear all,
I have the following problem: Consider an array of $N$ vectors $v_{i} \ i=1...N$ of size $L$ bits, where each bit is 1/0 with equal probability. I want to find a hash function $H()$ that ...

**12**

votes

**2**answers

375 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**10**

votes

**1**answer

531 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**15**

votes

**4**answers

752 views

### Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem:
Show, for all integers $1 \leq i \leq k$, that the univariate ...

**8**

votes

**0**answers

477 views

### Recent Fast Multiplication Algorithms for Large Integers

The STOC 2008 paper "Fast Integer Multiplication using Modular Arithmetic" by De et al
http://arxiv.org/abs/0801.1416 shows how to use $p$-adic numbers instead of $\mathbb C$ used in Furer's ...

**2**

votes

**2**answers

198 views

### How can I get all the good items using quantum search algorithm?

One can get a superposition of all good item using quantum search algorithm in $O$($\sqrt{N}$ ) time, but how one can get all the good items using quantum search algorithm?
I found that all the good ...

**1**

vote

**1**answer

181 views

### Special case of testing integer polynomials for irreducibility

How much easier is testing polynomials of the form x^n + ax + b for irreducibility (in Z[x]) than testing polynomials in general? I am especially interested in the case where n is prime, which may be ...

**5**

votes

**1**answer

570 views

### How much does a quantum oracle to find a needle in a haystack really cost?

Among the basic algorithms of quantum computations Lov Grover's result on quantum search stands out, both in regards to its intrinsic interest, and for its undisputable elegance.
Grover's algorithm ...

**4**

votes

**0**answers

153 views

### Differential operators that preserve real-rootedness

Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of ...

**3**

votes

**2**answers

163 views

### Determination of rationality and computing a rational parametrization

Suppose I have a hypersurface in $\mathbb{C}P^n$ given by some $f(z_1, \dots, z_{n+1}) = 0.$ Is there an algorithm which returns a rational parametrization if there is one, and "not rational" ...

**1**

vote

**2**answers

136 views

### Approximate version of a balanced incomplete block diagram

Let $S$ be a set of some size $n$. I'm interested in knowing about combinatorial designs that are approximately balanced incomplete block designs, that I want a collection of subsets $C$ of $S$ such ...

**16**

votes

**1**answer

2k views

### How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...

**3**

votes

**3**answers

405 views

### Are there any neat algorithm to factor a homogenous polynomial, given we know this polynomial factors into linear forms?

Are there any neat algorithm to factor a multivariable (more than 2 variables) homogenous polynomial, given we know this polynomial factors into linear forms?

**1**

vote

**0**answers

76 views

### Deciding / Approximating Parity of Small Depth Decision Trees

Let C be a circuit such that:
C: $\{0,1\}^n$ to $\{0,1\}$
the top most gate is a parity gate
all the inputs to the parity gate are small depth decision trees
there is a total of $2^{ log^k n}$ ...

**5**

votes

**0**answers

241 views

### Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...

**1**

vote

**2**answers

170 views

### Simultaneous Smith Normalization of a Composable Matrix Sequence

Let $\mathsf{R}$ be a PID and consider a collection of free, finitely generated $\mathsf{R}$-modules $V_1,\ldots,V_n$ along with module maps $m_j:V_j \to V_{j+1}$. That is, we have the following ...

**2**

votes

**1**answer

2k views

### FFT and Butterfly Diagram

Wikipedia presents butterfly as "a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into ...

**2**

votes

**0**answers

114 views

### Additional Constraint Baum Welch for HMMs

I'm trying to derive a special form of the Baum Welch algorithm where there is an additional constraint that the sum of emission probabilities over all states sums to one for each output symbol. ...

**1**

vote

**3**answers

1k views

### How to get the largest subset of a set of sets of intervals with no overlapping intervals

Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input ...

**0**

votes

**0**answers

172 views

### Maximum subset of set of Integers with minimum distance

Hi, i have a set of integers for example: {0,1,3,100,102} and i am looking for a maximum subset in which all elements have a minimum distance to all elements (or the "next" doesnt matter i guess) for ...

**2**

votes

**3**answers

269 views

### Generating a set of integer passwords that can be securely authenticated

First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
My question is as follows.
Given a positive integer $k$, determine a set of properties ...

**10**

votes

**1**answer

232 views

### Network flows with capacities on pairs of edges

Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...

**2**

votes

**1**answer

101 views

### A Parameter related to fractional chromatic number and Kneser Graphs

Let $t \gt 0$ be an integer and $G$ is a simple graph with $\chi_f(G) = t$. Then $t$= inf $\{ \frac{n}{k}| G \rightarrow KG(n,k)\}$ where $KG(n,k)$ is the Kneser graph.
Does there exist a $k$ such ...