Questions tagged [decomposition]
17 questions
5
votes
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Global optimization of state assignments in a directed graph with a tree-based distance cost
I am exploring a general optimization framework to solve problems characterized by the following structure. Any literature references, search terms, or algorithmic strategies would be greatly appreciated. I aim to unify several algorithms I devised…
Rolf Rolles
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4
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1 answer
One-dimensional packing problem: Optimal decomposition of music structure
I am currently working on my Master thesis on the visualization of music structure and I'm looking to find an optimal description of repetitions found in a piece of music.
Problem Description
Given a section range in a song in seconds (or samples) ,…
Job Savelsberg
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4
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1 answer
Term for a graph decomposition based on a maximum matching
Let $M$ be a maximum cardinality matching in a bipartite graph $G(X+Y,E)$. Let $X_0$ be the subset of $X$ unmatched by $M$. Define the following sequence:
$Y_1 = $ the neighbors of $X_0$ using edges in $E\setminus M$.
$X_1 = $ the neighbors of…
Erel Segal-Halevi
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3
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1 answer
Applying SVD compression to integral point images
Suppose that we have an $m\times n$ matrix $A$ of rank $n$, whose entries are 8-bit unsigned integers obtained from a grayscale image. Now we want to apply SVD to $A$ and to use the first $k$ singular values to construct the the best rank-$k$…
Peradventure
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1
vote
0 answers
Polygon decomposition into minimum star-shaped polygons
As the title suggests, I'm trying to implement an algorithm to decompose a polygon into the minimum number of star-shaped polygons. I've been searching for quite some time but I can't find any algorithms or even anything theoretical to study on this…
John Katsantas
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1 answer
What is the bit complexity of Gaussian eliminaton over $\Bbb F_q$?
Given matrix $M\in\Bbb F_q^{n\times n}$ with rank $r$ what is the complexity of converting to row-echelon form?
Is it $O(n^3\log q)$ or $O(n^3q)$ bit operations?
Technically $O(n^3)$ row operations in $O(\log q)$ bit words should be $O(n^3(\log…
Turbo
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Computational Considerations on least squares problems
I am reading Elements of Statistical Learning and read the following claim from the text (page 93, Chapter 3.7):
Least squares fitting is usually done via the Cholesky decomposition of the matrix $\mathbf{X}^T\mathbf{X}$ or a QR Decomposition of…
cgo
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1
vote
1 answer
Parallel algorithm for LU-decomposition
I need to implement LU-decomposition in Kaira. In Kaira the programmer writes the "parallel part" as the diagram similar to Petri Nets.
So, could you, please, recommend me some parallel algorithms for LU-decomposition which are really easy to…
Eenoku
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1 answer
Learning eigenvalue decomposition
How would you build a fully connected neural network that learns eigenvalue decomposition efficiently?
I wanted to build NNs that can predict certain properties about matrices which are NP-hard to compute but might require eigenvalue decomposition.…
arjo
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Relationships between path width and clique size of interval graphs
I faced the following claim on wikiepdia about interval graphs (https://en.wikipedia.org/wiki/Interval_graph):
The pathwidth of an interval graph is one less than the size of its maximum clique.
I have thought a while about it, but don't come up…
user20519169
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1
vote
1 answer
Converting a Directed Acyclic Graph to a Directed Tree
I'm wondering if anyone can help me with this. Say I have a DAG, I understand that it has no directed cycles, but it can have loops ( "diamonds" ).
My question is, is there a known way to convert such graph to a directed tree that has no loops. I…
T-Tory
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Decomposition of a bipartite graph into complete bipartite graphs by adding the smallest number of edges
A bipartite graph $G$ and an integer $K$ is given. I want to decompose $G$ into $K$ complete bipartite graphs by adding the smallest number of edges.
Below is an example of decomposition when $K=2$. The red line indicates an added edge.
Is there…
kivantium
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0
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1 answer
Fine-Grain parallel algorithm for LU-decomposition
How would you understand this pseudocode of parallel algorithm for LU-decomposition ? I'm confused mostly with the
min(i; j) - 1,
because I have no idea, what the author wanted to say by it. I know that it means " choose the lesser number of 'i'…
Eenoku
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0
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BCNF Decomposition: Confusion regarding given answer
I encountered the following question:
Given a relation R(A, B, C, D) with the following functional dependencies:
A -> B, C -> D, B -> C.
The BCNF Decomposition of R is:
A) {(A,B), (C,D), (B,C)}
B) {(A,B), (C,D), (A,C)}
C) {(B,C), (A,D), (A,B)}
D)…
Abhishek Bansal
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0
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Integer decomposition algorithm
Suppose I have a 32-bit integer $x$, I want to find $\{ x_i \}_{i \in 1\dots\ell}$ such that
$x = e + \sum_{i=1}^\ell x_i \cdot 2^{32 - B\cdot i}$
where the error $e$ is as small as possible. The parameter $\ell$ is the level of the decomposition…
lamba
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