Results 1  10
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20
The Union of Balls and its Dual Shape
, 1993
"... Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived f ..."
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Cited by 172 (12 self)
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Efficient algorithms are described for compuiing topological, combinatorial, and metric properties of ihe union of finitely many balls in R^d. These algorithms are based on a simplicial complex dual to a certain decomposition of the union of balls, and on short inclusionexclusion formulas derived from this complex. The algorithms are most relevant in R’3 where unions of finitely many balls are commonly used as models of molecules.
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 145 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
An Incremental Algorithm for Betti Numbers of Simplicial Complexes
, 1993
"... A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!i ..."
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Cited by 110 (14 self)
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A general and direct method for computing the betti numbers of the homology groups of a finite simplicial complex is given. For subcomplexes of a triangulation of S³ this method has implementations that run in time 0(’na(n)) and O(n), where n is the number of simplices in the triangulation. If app!ied to the family of ashapes of a finite point set in R³ ittakes time O(ncz(n)) to compute the betti numbers of all crshapes.
Computing Betti Numbers via Combinatorial Laplacians
 ALGORITHMICA
, 1998
"... We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, ..."
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Cited by 48 (1 self)
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We use the Laplacian and power method to compute Betti numbers of simplicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, ”, of eigenvalues which we have yet to understand fully. We numerically verify a conjecture of Björner, Lovász, Vrećica, and ˘Zivaljevic ́ on the chessboard complexes C.4; 6/, C.5; 7/, and C.5; 8/. Our verification suffers a technical weakness, which can be overcome in various ways; we do so for C.4; 6 / and C.5; 8/, giving a completely rigorous (computer) proof of the conjecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.
Coreduction homology algorithm
 Discrete & Computational Geometry
"... Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, ..."
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Cited by 29 (9 self)
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Abstract. A new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes, particularly strong for low dimensional sets embedded in high dimensions, is presented. The algorithm runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. 1.
Homology algorithm based on acyclic subspace
"... We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace ma ..."
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Cited by 22 (8 self)
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We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces the amount of data that needs to be processed in the algebraic way, and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms.
COREDUCTION HOMOLOGY ALGORITHM FOR INCLUSIONS AND PERSISTENT HOMOLOGY
"... Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the p ..."
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Cited by 15 (4 self)
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Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to twosided filtrations. In addition to describing the theoretical background, we present results of numerical experiments, as well as several applications to concrete problems in materials science. 1.
Recognizing badly presented Zmodules
 LINEAR ALGEBRA APPL
, 1993
"... Finitely generated Zmodules have canonical decompositions. When such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith ..."
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Cited by 11 (2 self)
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Finitely generated Zmodules have canonical decompositions. When such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith normal form computation, and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups. We investigate such issues as congruential techniques, sparsity considerations, pivoting strategies for GaussJordan elimination, lattice basis reduction and computational complexity. Our results, which are primarily empirical, show dramatically improved performance on previous methods.
Computing invariants of simplicial manifolds
, 2004
"... Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic ..."
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Cited by 8 (3 self)
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Abstract. This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book “Triangulated Manifolds with Few Vertices ” by Frank H. Lutz. The purpose of this chapter is to survey what is known about algorithms for the computation of algebraic invariants of topological spaces. Primarily, we use finite simplicial complexes as our model of topological spaces; for a discussion of different views see Section 4. On the way we give explicit definitions or constructions of all invariants presented. Note that we did not try to phrase all the results in their greatest generality. Similarly, we focus on invariants for which actual implementations exist. The reader is referred to Bredon’s monograph [2] for the wider perspective. For a related survey see Vegter [44]. 1. Homology
Surface Triangulation: A Survey
, 1996
"... This paper presents a brief survey of some problems and solutions related to the triangulation of surfaces. A surface (a two dimensional manifold, in the context of this paper) can be represented as a three dimensional function on a planar disk. In that sense, the triangulation of the disk induces a ..."
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Cited by 4 (0 self)
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This paper presents a brief survey of some problems and solutions related to the triangulation of surfaces. A surface (a two dimensional manifold, in the context of this paper) can be represented as a three dimensional function on a planar disk. In that sense, the triangulation of the disk induces a triangulation of the surface. Hence the emphasis of this paper is on triangulation on a plane. Apart from the issues in triangulation, this survey talks about the known upper and lower bounds on various triangulation problems. It is intended as a broad compilation of known results rather than an intensive treatise, and the details of most algorithms are skipped. 1 Introduction This survey assumes familiarity with the fundamental concepts of computational geometry. We define the triangulation problem as follows: Input: i. A set S of points, fp i g, such that each p i lies on the surface ii. A set of conditions, fC i g Output: A set S 0 of triples f(p i 1 ; p i 2 ; p i 3 )g such that e...