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Questions tagged [graph-theory]

A field of combinatorics relating to the study of vertices and the edges that connect them

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3
votes
2answers
96 views

Find shortest path around a cylinder represented by 3d triangular mesh

Suppose I have a 3d triangular mesh with the topology of a finite cylinder. Let $C$ be a vertex on that mesh. How can I find the shortest path from $C$ to itself that goes around the cylinder? By ...
0
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0answers
21 views

Controllability - Maximum Matching

I found this image on wikipedia referring to Barabasi's work on Network Controllability. I tried to verify it. We have a A matrix of dimensions (20 by 20) made as the image suggests. According to the '...
1
vote
1answer
194 views

Efficiently generate a random subgraph (Gs) with maximum degree K, using only edges from an existing graph G

I am looking find a way of efficiently generating a random, undirected subgraph $G_s$ with $N$ vertices, using a subset of edges from an exisiting undirected graph $G$, also of size $N$, where the ...
1
vote
0answers
16 views

Find all recurring subgraphs/patterns of maximal size in a single undirected, labeled, connected graph

I would like to identify all subgraphs of maximal size (maximum number of nodes) that are recurrent in a single undirected, labeled, connected graph. I provide exemples of input and expected output ...
4
votes
2answers
110 views

Efficient algorithm to decide if a graph is a cactus?

A cactus is a connected graph in which every edge belongs to at most one simple cycle. How should one modify the Depth First Search algorithm to obtain an efficient algorithm that determines if a ...
1
vote
1answer
49 views

Vehicle Route assignment with capacity constraint

Problem Background I'm trying to find a solution/model to the following problem: Let's consider a cellular network (mobile network, ie., hexagonal cells) denoted $N$ composed of $|N|$ cells. Each ...
0
votes
1answer
117 views

Finding a shortest path in a graph

If each edge of a graph $G$ is unweighted or has equal weights, then the shortest path between two nodes in that graph is the path that contains the fewest number of edges. Such a path can be obtained ...
11
votes
3answers
813 views

Testing if two 12x12 matrices have the same determinant

I am given a $12 \times 12$ matrix $Q$ that is symmetric, invertible, positive definite and dense. I need to test if $$\det(Q) = \det(12I-Q-J) \; \; (1)$$ where $J$ is the all ones matrix. I am ...
0
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1answer
157 views

How to reorder/cluster adjacency matrix to maximize the interaction along the super diagonal?

I have the following code which takes a DataFrame and plot the pdist matrix. ...
3
votes
0answers
103 views

Factorize laplacian in terms of first derivative matrix

I am trying to factorize the following Laplacian matrix in terms of $ D^TD$, D is the first derivative matrix. The tridiagonal form of the secon derivative matrix using Neumann boundary condition is ...
1
vote
1answer
77 views

Splittable and non-splittable flows in the network flow problem

I am working on a multi-commodity flow problem where for a graph $G=(V, E)$, some flows are permitted to be split and some flows should strictly follow one path. I have formulated this problem as ...
14
votes
6answers
974 views

Approximate spectrum of a large matrix

I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard. I am willing to settle for an approximation. Are there approximation ...
4
votes
2answers
190 views

Moore-Penrose pseudoinverse of singular rank degenerate matrix

I am trying to attain the Moore-Penrose pseudoinverse of a very large, very sparse, rank-degenerate, singular, and square matrix. ($75000 \times 75000$, near rank). The matrix is a graph Laplacian and ...
3
votes
1answer
38 views

Software for finding a minimum vertex cover for a hypergraph

A hypergraph $H = (V,E)$ consists of a finite set of vertices, say $V=\{1, \dots, n\}$ and a set of hyperedges $E \subseteq \mathcal{P}(V)$. We call $H$ a $k$-hypergraph if all $|e| = k$ for all $e\in ...
0
votes
1answer
214 views

Hypergraph matching -> adjacency matrix?

I need to do a matching on a hypergraph. I read that in the case of a hypergraph there is no adjacency matrix. How do I represent edges then?
2
votes
2answers
150 views

Find connected circles

I have a problem as follows: We have a set of circles (we know the radius r and the center point c in Rd of each circle) We ...
13
votes
6answers
4k views

What is a common file/data format for a mesh (for FEM)?

I'm developing an FEM simulation. For early testing, I will use simple self-written mesher and visualisation of the mesh graph. But I want to prepare my program to use data generated by an existing ...
0
votes
1answer
61 views

simple and fast graph-clustering for paralelization of finite element simulations

I'm learning to use OpenCL to optimize some of my simulations. I realized that I need some sort of Graph-clustering or graph-partitioning to exploit efficiently local memory for un-ordered meshes. ...
1
vote
0answers
182 views

First approximation to the TSP in a non-complete Graph

I'm trying to solve the Travelling Salesman Problem in a non-complete graph $(G,E)$ using genetic algorithms. My problem is that I can't find a good first approximation by the usual greedy algorithms,...
2
votes
2answers
410 views

PageRank using Inverse Iteration Method by Cleve Moler

I was trying to understand how to use the inverse interation method to compute the page rank as an exercise. In this chapter (page 4) about page rank (by Cleve Moler), the author suggests to use the ...
0
votes
1answer
574 views

Can a Depth first traversal of a graph visit node more than once? [closed]

Is there a way in which Depth First Traversal will put a node into the stack more than once using the general algorithm as shown here. Also, is it compulsory for all nodes to be entered into the ...
1
vote
1answer
63 views

Does the box-covering algorithm work also for directed graphs?

According to this article from Wikipedia, the box-covering algorithm calculates the fractal dimension of a graph. The algorithm is based on the concept of distance between nodes; see for example the ...
1
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0answers
26 views

Chinese character component-based layout engine algorithms [closed]

I am looking for a PDF describing Chinese character component-based layout engine algorithms. These should allow for recursive encodings of radical and non-radical components, and the encodings should ...
1
vote
0answers
119 views

Updating factorization of Laplacian (add/remove edges)

For a graph $G=(V,E)$, recall that the unweighted Laplacian is $L:=D^\top D$, where $D\in\{-1,0,1\}^{|E|\times|V|}$ is the graph "gradient" operator that subtracts adjacent vertex values onto edges. ...
6
votes
1answer
7k views

How to implement a dynamic programming solution to the 2D bitonic euclidean traveling salesman problem?

I understand that a bitonic tour crosses all vertices with one monotonic path traveling from the left most point to the right most point, then monotonically from the left most point to the right most ...
1
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0answers
43 views

Alternative to two “for” loops in finding best neighborhoods for TSP?

I am trying to solve Travelling Salesman Problems using tabu search. I have been able to successfully find "near enough" optimal solutions (as well as one optimal, yay!). For the moment I am using ...
1
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0answers
40 views

Interior nodes of a closed graph?

Does anybody know if any graph partitioner library such as Metis, Scotch, or Zoltan can (besides splitting a domain), differentiate between internal (i) and boundary (b) nodes?
5
votes
1answer
69 views

Optimal partitioning of a graph

Consider a planar graph, where each node is associated with a weight. I would like to partition the graph such that the sum of the node weights in each group satisfy a minimum requirement. However, I ...
3
votes
1answer
103 views

Sparse Matrix Reordering

Matrix reorderings are important for many direct solvers. Sometimes the objective is to reduce the bandwith or the generated fill in by LU Decomposition. I am interested in a reordering which reduces ...
4
votes
0answers
79 views

Graph optimization for parallel processing

Consider the following example structure of overlapping images marked A,B,C,D. The possible overlaps are marked by gray color: The structure can be represented by a weighted undirected graph (images ...
4
votes
3answers
567 views

Minimization of non-linear function

Problem Summary I am trying to estimate the (x,y) coordinates of each node in a graph, where I know the distances between connected nodes. For example Given this ...
1
vote
1answer
121 views

Measure the differences in vertices density in a graph?

Lets say in Graph $G$ we have two vertices $v$ and $u$, each vertex is connected to several neighbors by edges describing the distances $d_{ij}$ from these neighbors. The neighbors themselves are also ...
3
votes
2answers
613 views

How to calculate efficiently mesh edges midpoints?

I have a 2D mesh of triangles used in Finite Element method to discretize the domain. I want to calculate the midpoints of all the edges because I want to use $\mathbb{P}^2$ elements. I am using ...
1
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1answer
64 views

Finding Common Side of Triangle

Given a triangulation (geometry), are there known algorithm in finding common side of triangles, that is O(N) or better?
0
votes
1answer
139 views

how to partition a graph(matrix) into subdomains with different sizes

i am studying the solver for PageRank problems which drived from the web link graph. I have tried using METIS to divided the matrix into subdomains, but METIS can only produce subdomains with nearly ...
3
votes
0answers
67 views

Absorbing BC's / PML on a graph

The wave equation, $$\ddot{u} = c^2 \Delta u,$$ can be generalized to abstract graphs by using the negative graph Laplacian in place of the physical Laplacian. Is there a graph-theoretic analog of ...
1
vote
1answer
666 views

Graph drawing with constraints

I have a (cyclic undirected) graph stored in a suitable structure (which depends on the language chosen). Nodes represent places in the real world and edges represent connection between them, so I ...
0
votes
2answers
360 views

How to represent molecules and compare equality

I originally asked this question at StackOverflow, and was suggested to bring it here. I've seen this question about the representation of molecules in memory, and it makes sense to me (tl;dr ...
1
vote
1answer
1k views

Efficient way to compute the cumulative weights of all subtrees rooted at each node in a tree?

I have a tree data structure (rooted, unbalanced, with unbounded branching factor), where each individual node has an associated 'weight'. For every node $n$ in the tree, I'd like to compute the ...
2
votes
1answer
84 views

graph theory operations to explore structure of a graph

I'm analyzing experimental data, and I've produced graphs that look similar to the attached sketch. When I look at the graph, I see structure in the connections (connections tend to be local, there is ...
2
votes
0answers
89 views

Algorithm for optimizing graph interconnectivity

I have a partiuclar kind of graph problem and (not having a background in graph algorithms) I would like to know how this kind of problem is called in the literature and what algorithms exist for ...
1
vote
2answers
150 views

Rearrange a dense distance matrix to a 2x2 non-perfect block diagonal form

I have a distance matrix (square, symmetrical, non-negative, dense). I want to split the objects into two well-connected groups. Mathematically speaking, I want to group (re-arrange) the rows/columns ...
2
votes
1answer
188 views

Algorithm to equalize the area of random tessellation of various polygons

I am looking for an algorithm that I can apply for a random tessellation of polygons with different areas. The algorithm can relax the geometry of the polygons to a condition that all of them would ...
11
votes
3answers
6k views

I am looking for a parallel dynamic graph library in C++

Hello scicomp community, I have worked in the area of graph algorithms using frameworks such as NetworkX (Python), JUNG and YFiles (Java). I am now entering the area of parallel and high perfomance ...
2
votes
1answer
106 views

k-splittable flow problem

The maximum k-splittable s-t flow problem(MkSF) that aims to find a maximum k-splittable flow between a given source and sink node is NP-hard. We do not require the paths to be disjoint, not even ...
3
votes
0answers
128 views

The traveling salesman problem - Using Space Renormalization

Image attached is where I am at the moment. Blue dots=points/cities, Black x's represent central points in each box that contains at least one city, and pink dots represent the midpoint of these ...
0
votes
1answer
646 views

sparse matrix format with fast row and column access

Is there an efficient storage format for general, non-symmetric sparse matrices for which one can find all non-zero entries in a given row or column in $O(d)$ time? ($d$ is the max number of non-zero ...
6
votes
1answer
11k views

Concave polygon 'hull' finding

I implemented an algorithm to find the alpha shape of a set of points. The alpha shape is a concave hull for a set of points, whose shape depends on a parameter alpha deciding which points make up the ...
3
votes
1answer
68 views

Minimizing expression DAGs

Say I have an expression involving adds, subtractions, and multiplications. I know that it is safe to assume commutativity, associativity, distributivity, etc., and would like to automatically ...
3
votes
1answer
251 views

Optimal Scheduling of Parallel Tasks with Known Dependencies

This is maybe a trivial question, but I am stuck with the problem. Suppose we have a general graph: $$G=(V,E)$$ Each edge represents a task, each vertex represents a data for the task (hence each ...