952 research outputs found
An Efficient Representation for Filtrations of Simplicial Complexes
A filtration over a simplicial complex is an ordering of the simplices of
such that all prefixes in the ordering are subcomplexes of . Filtrations
are at the core of Persistent Homology, a major tool in Topological Data
Analysis. In order to represent the filtration of a simplicial complex, the
entire filtration can be appended to any data structure that explicitly stores
all the simplices of the complex such as the Hasse diagram or the recently
introduced Simplex Tree [Algorithmica '14]. However, with the popularity of
various computational methods that need to handle simplicial complexes, and
with the rapidly increasing size of the complexes, the task of finding a
compact data structure that can still support efficient queries is of great
interest.
In this paper, we propose a new data structure called the Critical Simplex
Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica
'17]. Our data structure allows one to store in a compact way the filtration of
a simplicial complex, and allows for the efficient implementation of a large
range of basic operations. Moreover, we prove that our data structure is
essentially optimal with respect to the requisite storage space. Finally, we
show that the CSD representation admits fast construction algorithms for Flag
complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
Motion Planning of Legged Robots
We study the problem of computing the free space F of a simple legged robot
called the spider robot. The body of this robot is a single point and the legs
are attached to the body. The robot is subject to two constraints: each leg has
a maximal extension R (accessibility constraint) and the body of the robot must
lie above the convex hull of its feet (stability constraint). Moreover, the
robot can only put its feet on some regions, called the foothold regions. The
free space F is the set of positions of the body of the robot such that there
exists a set of accessible footholds for which the robot is stable. We present
an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n))
space for n discrete point footholds where alpha(n) is an extremely slowly
growing function (alpha(n) <= 3 for any practical value of n). We also present
an algorithm for computing F when the foothold regions are pairwise disjoint
polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log
n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly
growing function). These results are close to optimal since Omega(n2) is a
lower bound for the size of F.Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE
Robotics & Automation 9
Constructing Intrinsic Delaunay Triangulations of Submanifolds
We describe an algorithm to construct an intrinsic Delaunay triangulation of
a smooth closed submanifold of Euclidean space. Using results established in a
companion paper on the stability of Delaunay triangulations on -generic
point sets, we establish sampling criteria which ensure that the intrinsic
Delaunay complex coincides with the restricted Delaunay complex and also with
the recently introduced tangential Delaunay complex. The algorithm generates a
point set that meets the required criteria while the tangential complex is
being constructed. In this way the computation of geodesic distances is
avoided, the runtime is only linearly dependent on the ambient dimension, and
the Delaunay complexes are guaranteed to be triangulations of the manifold
Building Efficient and Compact Data Structures for Simplicial Complexes
The Simplex Tree (ST) is a recently introduced data structure that can
represent abstract simplicial complexes of any dimension and allows efficient
implementation of a large range of basic operations on simplicial complexes. In
this paper, we show how to optimally compress the Simplex Tree while retaining
its functionalities. In addition, we propose two new data structures called the
Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the
compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List
under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201
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