521 research outputs found
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
This work introduces a number of algebraic topology approaches, such as
multicomponent persistent homology, multi-level persistent homology and
electrostatic persistence for the representation, characterization, and
description of small molecules and biomolecular complexes. Multicomponent
persistent homology retains critical chemical and biological information during
the topological simplification of biomolecular geometric complexity.
Multi-level persistent homology enables a tailored topological description of
inter- and/or intra-molecular interactions of interest. Electrostatic
persistence incorporates partial charge information into topological
invariants. These topological methods are paired with Wasserstein distance to
characterize similarities between molecules and are further integrated with a
variety of machine learning algorithms, including k-nearest neighbors, ensemble
of trees, and deep convolutional neural networks, to manifest their descriptive
and predictive powers for chemical and biological problems. Extensive numerical
experiments involving more than 4,000 protein-ligand complexes from the PDBBind
database and near 100,000 ligands and decoys in the DUD database are performed
to test respectively the scoring power and the virtual screening power of the
proposed topological approaches. It is demonstrated that the present approaches
outperform the modern machine learning based methods in protein-ligand binding
affinity predictions and ligand-decoy discrimination
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are
common objects in the computer graphics or topological data analysis
literature. Defining good metrics between these objects has become an important
question for applications, where it matters to quantify the extent by which two
given Reeb graphs differ. Recent contributions emphasize this aspect, proposing
novel distances such as {\em functional distortion} or {\em interleaving} that
are provably more discriminative than the so-called {\em bottleneck distance},
being true metrics whereas the latter is only a pseudo-metric. Their main
drawback compared to the bottleneck distance is to be comparatively hard (if at
all possible) to evaluate. Here we take the opposite view on the problem and
show that the bottleneck distance is in fact good enough {\em locally}, in the
sense that it is able to discriminate a Reeb graph from any other Reeb graph in
a small enough neighborhood, as efficiently as the other metrics do. This
suggests considering the {\em intrinsic metrics} induced by these distances,
which turn out to be all {\em globally} equivalent. This novel viewpoint on the
study of Reeb graphs has a potential impact on applications, where one may not
only be interested in discriminating between data but also in interpolating
between them
Topological Hierarchies and Decomposition: From Clustering to Persistence
Hierarchical clustering is a class of algorithms commonly used in exploratory data analysis (EDA) and supervised learning. However, they suffer from some drawbacks, including the difficulty of interpreting the resulting dendrogram, arbitrariness in the choice of cut to obtain a flat clustering, and the lack of an obvious way of comparing individual clusters. In this dissertation, we develop the notion of a topological hierarchy on recursively-defined subsets of a metric space. We look to the field of topological data analysis (TDA) for the mathematical background to associate topological structures such as simplicial complexes and maps of covers to clusters in a hierarchy. Our main results include the definition of a novel hierarchical algorithm for constructing a topological hierarchy, and an implementation of the MAPPER algorithm and our topological hierarchies in pure Python code as well as a web app dashboard for exploratory data analysis. We show that the algorithm scales well to high-dimensional data due to the use of dimensionality reduction in most TDA methods, and analyze the worst-case time complexity of MAPPER and our hierarchical decomposition algorithm. Finally, we give a use case for exploratory data analysis with our techniques
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
The Homogeneous Broadcast Problem in Narrow and Wide Strips
Let be a set of nodes in a wireless network, where each node is modeled
as a point in the plane, and let be a given source node. Each node
can transmit information to all other nodes within unit distance, provided
is activated. The (homogeneous) broadcast problem is to activate a minimum
number of nodes such that in the resulting directed communication graph, the
source can reach any other node. We study the complexity of the regular and
the hop-bounded version of the problem (in the latter, must be able to
reach every node within a specified number of hops), with the restriction that
all points lie inside a strip of width . We almost completely characterize
the complexity of both the regular and the hop-bounded versions as a function
of the strip width .Comment: 50 pages, WADS 2017 submissio
- …