17 research outputs found
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
New error measures and methods for realizing protein graphs from distance data
The interval Distance Geometry Problem (iDGP) consists in finding a
realization in of a simple undirected graph with
nonnegative intervals assigned to the edges in such a way that, for each edge,
the Euclidean distance between the realization of the adjacent vertices is
within the edge interval bounds. In this paper, we focus on the application to
the conformation of proteins in space, which is a basic step in determining
protein function: given interval estimations of some of the inter-atomic
distances, find their shape. Among different families of methods for
accomplishing this task, we look at mathematical programming based methods,
which are well suited for dealing with intervals. The basic question we want to
answer is: what is the best such method for the problem? The most meaningful
error measure for evaluating solution quality is the coordinate root mean
square deviation. We first introduce a new error measure which addresses a
particular feature of protein backbones, i.e. many partial reflections also
yield acceptable backbones. We then present a set of new and existing quadratic
and semidefinite programming formulations of this problem, and a set of new and
existing methods for solving these formulations. Finally, we perform a
computational evaluation of all the feasible solverformulation combinations
according to new and existing error measures, finding that the best methodology
is a new heuristic method based on multiplicative weights updates
On the maximal number of real embeddings of spatial minimally rigid graphs
The number of embeddings of minimally rigid graphs in is (by
definition) finite, modulo rigid transformations, for every generic choice of
edge lengths. Even though various approaches have been proposed to compute it,
the gap between upper and lower bounds is still enormous. Specific values and
its asymptotic behavior are major and fascinating open problems in rigidity
theory. Our work considers the maximal number of real embeddings of minimally
rigid graphs in . We modify a commonly used parametric
semi-algebraic formulation that exploits the Cayley-Menger determinant to
minimize the {\em a priori} number of complex embeddings, where the parameters
correspond to edge lengths. To cope with the huge dimension of the parameter
space and find specializations of the parameters that maximize the number of
real embeddings, we introduce a method based on coupler curves that makes the
sampling feasible for spatial minimally rigid graphs.
Our methodology results in the first full classification of the number of
real embeddings of graphs with 7 vertices in , which was the
smallest open case. Building on this and certain 8-vertex graphs, we improve
the previously known general lower bound on the maximum number of real
embeddings in
An algorithm to enumerate all possible protein conformations verifying a set of distance constraints
Realizing Euclidean distance matrices by sphere intersection
International audienceThis paper presents the theoretical properties of an algorithm to find a realization of a (full) n × n Euclidean distance matrix in the smallest possible embedding dimension. Our algorithm performs linearly in n, and quadratically in the minimum embedding dimension, which is an improvement w.r.t. other algorithms
Symmetries in distance geometry
Orientadores: Fernando Eduardo Torres Orihuela, Carlile Campos LavorDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O problema principal de Geometria de Distâncias (GD) é determinar as posições de um conjunto de pontos, considerando conhecidas algumas distâncias. Este trabalho tem como principal objetivo estudar uma classe particular de problemas da GD, onde aparecem simetrias que podem ajudar a caracterizar todas as soluções do problema. Apresentamos os resultados para o R3, os quais podem ser generalizados para o RnAbstract: The main problem of Distance Geometry (GD) is to determine the positions of a set of points, considering some known distances. This work has as main objective to study a particular class of problems of DG, where there are simmetries that can be used to characterize all the solutions of the problem. We present the results for R3, which can be also generalized to RnMestradoMatematicaMestre em MatemáticaCAPE