All Questions
Tagged with ds.algorithms linear-algebra
62 questions
0
votes
1
answer
206
views
Construction of a collection of subsets of $\{1,2,\ldots,n\}$ with certain properties
Let $n$ be a large positive integer. Given a collection $\mathfrak S$ of subsets of $[n] := \{1,2,\ldots,n\}$, and a vector $z=(z_1,\ldots,z_n)\in \{\pm 1\}^n$, define
$$
f_{\mathfrak S}(z) := \sum_{\...
- 291
3
votes
2
answers
166
views
Worst-case complexity of computing a certain non-standard dot product + algorithms realizing this complexity
Let $n$ be a large positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an inner-product on $\mathbb R^n$ by
\begin{eqnarray}
\langle x,y\rangle_{\...
- 291
0
votes
1
answer
78
views
An inequality about median of points in higher dimensions
Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\...
- 1
3
votes
1
answer
235
views
An optimization problem
I am considering the following optimization problem. Let $P$ be a set of $n$ points in $\mathbb{R}^d$
maximize $\sum_{p\in P}\vert\langle \Vert p\Vert p, \hat{x}\rangle\vert$ subject to $\Vert\hat{x}\...
- 265
4
votes
0
answers
92
views
Complexity of Encoding a Matroid Flow Problem in a Matrix
Context:
Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$.
We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
- 696
1
vote
1
answer
346
views
What is the complexity of this submatrix selection problem?
We have a $kn\times kn$ matrix $M$ made of $n^2$ many $k\times k$ blocks.
We want to find an $n\times n$ submatrix such that each row and column is from distinct window of size $k$ such that the sum ...
- 13.3k
-3
votes
1
answer
145
views
Finding a non-negative solution to an integer system of linear equations
Let $A$ be an $n \times m$ integer matrix and consider the system of equations $Ax = b$ where $b$ is an integer vector. I want to find a solution $x$, assuming one exists, such that the components of $...
- 215
5
votes
0
answers
222
views
Is there a fast algorithm for inverting a sparse matrix?
I am doing research on a random-walk like problem. As a critical part of my solution, I need to invert a non-singular sparse matrix of size $n \times n$ and with $O(n)$ non-empty entries. I'm working ...
- 235
9
votes
3
answers
959
views
Non-Orthogonal Vectors Problem
Consider the following problems:
Orthogonal Vectors Problem
Input: A set $S$ of $n$ Boolean vectors each of length $d$.
Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
- 5,167
0
votes
1
answer
43
views
Decomposing outer product or general rank factorization over $\Bbb F_q$
Given matrix $M\in\Bbb F_q^{n\times n}$ with the promise that there are two matrices $A\in\Bbb F_q^{n\times 1}$ and $B\in\Bbb F_q^{1\times n}$ such that $AB=M$ is there a deterministic $O((n\log q)^c)$...
- 13.3k
-1
votes
1
answer
128
views
given a set of $n$ points in $d$-dimensional space and the basis vectors of some subspace, how to find all the points on that space?
given a set $A$ of $n$ points with integer coordinates in $\mathbb{R}^d$, and $k<d$ basis vectors of a subspace $K$ of $\mathbb{R}^d$, is there an efficient algorithm that returns all points from $...
7
votes
1
answer
192
views
Rank-robustness of the parallel complexity of linear algebra problems
It is known that most computational problems related to linear algebra
can be computed in $NC^2$ - i.e. for an $n\times n$ matrix $A$, over the reals
or a finite field, we can compute the rank of $A$, ...
- 1,224
1
vote
0
answers
126
views
Can we make a matrix stable by changing its upper-left submatrix?
A matrix $A$ is called strictly stable if its eigenvalues have negative real parts. Given a matrix $A \in \mathbb{R}^{n \times n}$, suppose we can change its upper-left $k \times k$ submatrix at will (...
- 11
7
votes
1
answer
275
views
Using an oracle to find a vector $b$ for which $Ax=b$ has a solution
There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
- 233
12
votes
2
answers
3k
views
Memory requirement for fast matrix multiplication
Suppose we want to multiply $n \times n$ matrices. The slow matrix multiplication algorithm runs in time $O(n^3)$ and uses $O(n^2)$ memory. The fastest matrix multiplication runs in time $n^{\omega + ...
- 3,548
6
votes
1
answer
225
views
Checking properties of matrices
Given a sparse matrix $A$ in $\mathbb{Z}^{n\times n}$, how easily could one check whether a coefficient $\alpha_k$ of the characteristic polynomial $P_A$ of $A$ is equal to $0$ (without the need to ...
- 73
1
vote
0
answers
187
views
How to efficiently generate a random 0-1 matrix of a given rank
How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
- 73
2
votes
1
answer
500
views
Matrix multiplication with transpose
Let $A,B\in\mathbb{F}^{n\times n}$ be two $n\times n$ matrices over
the underlying field $\mathbb{F}$. In addition, $A$ is guaranteed
to be a symmetric matrix, i.e, $A=A^{T}$. We assume complexity ...
- 727
6
votes
2
answers
2k
views
Min Hamming distance of a given string from substrings of another string
Let $U$ be a small finite set.
Consider the following problem:
Input: two strings $u \in U^k$ and $v\in U^n$ with $k \leq n$.
Output: a (contiguous) substring of $v$ of length $k$
with the minimum ...
- 113
2
votes
0
answers
266
views
Inclusion of polytopes
Consider the following two system of linear (in)eqaulities:
$S = Ax \leq b;\; Cx = e$
$T = Dx \leq d;\; Gx = g$
How can I check if $S\cap \neg T=\emptyset$ where $\neg T$ is the complement of the ...
- 253
1
vote
0
answers
87
views
Compute basis of vertex set of polytope
I am wondering whether there is an efficient algorithm to compute the basis of the set of vertices of a polytope.
Formally,
INPUT: a polytope
$$\Xi=\{(\vec{a}_1\vec{x}+\vec{b}_1, \cdots, \vec{a}_m\...
- 89
4
votes
1
answer
589
views
Rate of convergence for the Perron–Frobenius theorem
The Perron–Frobenius Theorem states the following.
Let $A = (a_{ij})$ be an $n \times n$ irreducible, non-negative matrix ($a_{ij} \geq 0, \forall i,j: 1\leq i,j \leq n$). Then the following ...
- 1,601
1
vote
0
answers
89
views
k closest points that belong to a set
This is a question from theory community, but I came across this issue in a practical problem. So just have this in mind.
I have a set of real vectors:
$$
S = \lbrace v_1, \dots, v_n \rbrace
$$
$...
- 749
1
vote
0
answers
118
views
Computing a sparse eigenvector
Given a matrix $A$ with distinct eigenvalues, can I find a sparsest eigenvector of it in polynomial time?
It is tempting to say that one can simply compute the eigenvectors and pick the sparsest ...
- 11
3
votes
0
answers
182
views
Algorithm (parallel and serial) for Gram-Schmidt
Suppose we are given $m$ vectors $v_1, \dots, v_m$ in $n$-dimensional space $\mathbf R^n$ (or perhaps they are specified up to $b$ bits of precision). I would like to find an orthonormal basis for the ...
- 3,548
2
votes
0
answers
283
views
Running time of Jacobi vs Golub-Kahan for SVD
For an $m \times n$ matrix, what is the running time for computing the Singular Value Decomposition (SVD for short) via
Jacobi's method, and
Golub-Kahan?
The source I read mentions that Jacobi's ...
- 121
7
votes
0
answers
578
views
An algorithm to compute the number of paths of length at most k
So I had to answer the following question:
Given a graph $G = (V, E)$, and two vertices $v_i, v_j$, compute the number of walks between $v_i$ and $v_j$ of length at most $k$. $G$ is not too large, ...
- 517
19
votes
4
answers
712
views
How to obtain the unknown values $a_i,b_j$ given an unordered list of $a_i-b_j\mod N$?
Can anyone help me with the following problem?
I want to find some values $a_i,b_j$ (mod $N$) where $i=1,2,…,K, j=1,2,…,K $ (for example $K=6$), given a list of $K^2$ values that correspond to the ...
- 193
22
votes
3
answers
798
views
Bigger picture behind the choice of matrices in the Strassen algorithm
In the Strassen algorithm, to compute the product of two matrices $\mathbf{A}$ and $\mathbf{B}$, the matrices $\mathbf{A}$ and $\mathbf{B}$ are divided into $2 \times 2$ block matrices and the ...
user4292
13
votes
0
answers
198
views
Complexity to compute the eigenvalue signs of the adjacency matrix
Let $A$ be the $n\times n$ adjacency matrix of a (non-bipartite) graph. Assume that we are given the amplitudes of its eigenvalues, i.e., $|\lambda_1|=a_1,\ldots, |\lambda_n|=a_n$, and we would like ...
- 1,356
2
votes
1
answer
402
views
All Pairs Shortest Path - Directed graph with integer weights
I don't understand how Distance Product works (or Min Plus Product). If we replace each argument in $A$ from $a_{i,j}$ to $x^{a_{i,j}}$ and each argument in $B$ from $b_{i,j}$ to $x^{b_{i,j}}$ and ...
- 241
1
vote
0
answers
203
views
Trace minimization with an orthogonality constraint
For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under the constraint:
that $X$ is orthogonal.
All the matrices have real entries and $A,B$ are ...
- 111
4
votes
0
answers
677
views
What about apply maxplus algebra for all-pairs shortest paths?
I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
10
votes
1
answer
427
views
What is the asymptotically fastest known algorithm for computing the nullspace of a matrix?
I know Gaussian Elimination takes $O(n^3)$ arithmetic operations, but I'm unsure if any better algorithms are known.
- 2,531
17
votes
2
answers
442
views
similar matrices
Given two $n \times n$ matrices $A$ and $B$, the problem of deciding if there exist a permutation matrix $P$ such that $B = P^{-1}AP$ is equivalent to GI(Graph ...
- 1,311
13
votes
1
answer
1k
views
Sampling from Multivariate Gaussian with Graph Laplacian (inverse) Covariance
We know from e.g. Koutis-Miller-Peng (based on work of Spielman & Teng), that we can very quickly solve linear systems $A x = b$ for matrices $A$ that are the graph Laplacian matrix for some ...
- 681
-5
votes
1
answer
320
views
Solving a system of linear inequations
Consider the following system of inequalities:
$Ax=b$;
$x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. a) How feasibility of this system can ...
- 253
9
votes
1
answer
643
views
Efficiently solve a system of strict linear inequalities with all coefficients equal to 1 without using a general LP solver?
Per the title, other than using a general purpose LP solver, is there an approach for solving systems of inequalities over variables $x_i, \ldots, x_k$ where inequalities have the form $\sum_{i \in I} ...
- 747
1
vote
0
answers
192
views
Complexity of checking whether linear equations have a positve solution [closed]
Consider a system of linear equations $Ax=0$, where $A$ is a $n\times n$ matrix with rational entries. Assume that the rank of $A$ is $<n$. What is the complexiy to check
whether it has a solution $...
- 109
11
votes
1
answer
489
views
Constructing vectors in general position
Let a real $k\times n$ ($k\le n$) matrix ${\bf A}$ with the property that any collection of $k$ columns is full rank.
Q: Is there an efficient way to deterministically find a vector ${\bf a}$ such ...
- 1,356
9
votes
2
answers
692
views
Midpoint solutions to linear programs
There is a linear program for which I want not merely a solution but a solution that's as central as possible on the face of the polytope that assumes the minimal value.
A priori, we expect the ...
- 1,226
19
votes
3
answers
2k
views
Determinant modulo m
What are the known efficient algorithms for computing a determinant of an integer matrix with coefficients in $\mathbb{Z}_m$, the ring of residues modulo $m$. The number $m$ may not be prime but ...
- 353
13
votes
1
answer
551
views
Algorithmic Vector Problem
I have an algebraic problem related to vectors in the field GF(2).
Let $v_1, v_2, \ldots, v_m$ be (0,1)-vectors of dimension $n$, and $m=n^{O(1)}$. Find a polynomial time algorithm that finds a (0,1)-...
- 674
7
votes
1
answer
275
views
determining if a matrix of linear forms represents a non-degenerate matrix
Let $k$ be a field with $p$ elements. Consider the following computational problem
Input: a natural number $n$, $n^2$ linear forms $M_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X_{11}, \ldots X_{...
- 301
16
votes
4
answers
4k
views
Definition of matrix-multiplication exponent $\omega$
Colloquially, the definition of the matrix-multiplication exponent $\omega$ is the smallest value for which there is a known $n^{\omega}$ matrix-multiplication algorithm. This is not acceptable as a ...
- 3,548
10
votes
1
answer
502
views
Finding a cutting plane that splits a polyhedron evenly
Say we have a polyhedron in standard form:
\begin{equation*}
\begin{array}{rl}
\mathbf{A}\mathbf{x} = \mathbf{b} \\\\
\mathbf{x} \ge 0
\end{array}
\end{equation*}
Are there any known methods for ...
3
votes
0
answers
332
views
Taking Square Roots of Matrices over Z/nZ
Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a ...
- 331
21
votes
1
answer
828
views
What is the most general structure on which matrix product verification can be done in $O(n^2)$ time?
In 1979, Freivalds showed that verifying matrix products over any field can be done in randomized $O(n^2)$ time. More formally, given three matrices A, B, and C, with entries from a field F, the ...
- 13.8k
14
votes
0
answers
633
views
Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)
Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same.
First, I will define a few ...
- 1,983
16
votes
2
answers
7k
views
What is the fastest algorithm to compute rank of a rectangular matrix?
Given an $m \times n$ matrix (assuming $m \ge n$), what is the fastest algorithm to compute its rank and basis of the columns?
I am aware it can be solved through linear matroid intersection, which ...
- 163