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30
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 109 (19 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.
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 39 (2 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
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 20 (8 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 17 (9 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.
An efficient algorithm for the transversal hypergraph generation
 Journal of Graph Algorithms and Applications
"... The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving t ..."
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Cited by 12 (0 self)
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The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration and, thus, its memory requirements are polynomially bounded to the size of the input hypergraph. Although no time bound for the algorithm is given, experimental evaluation and comparison with other approaches have shown that it behaves well in practice and it can successfully handle large problem instances.
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 12 (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.
Computing Homology Group Generators of Images Using Irregular Graph Pyramids
, 2006
"... We introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure i.e. an irregular graph pyramid. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph p ..."
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Cited by 6 (4 self)
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We introduce a method for computing homology groups and their generators of a 2D image, using a hierarchical structure i.e. an irregular graph pyramid. Instead of computing homology generators in the base where the number of entities (cells) is large, we first reduce the number of cells by a graph pyramid. Then homology generators are computed efficiently on the top level of the pyramid, since the number of cells is small, and a top down process is then used to deduce homology generators in any level of the pyramid, including the base level i.e. the initial image. We show that the new method produces valid homology generators and present some experimental results. In this report we also show that the generators of the first homology groups of a 2D image, computed with this pyramid based method always fit on the borders of the regions.
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 6 (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
Morse theory for filtrations and efficient computation of persistent homology, in preparation
"... Abstract. We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. ..."
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Cited by 5 (1 self)
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Abstract. We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
Topological Lower Bounds for the Chromatic Number: A Hierarchy
 JAHRESBER. DEUTSCH. MATH.VEREIN
, 2004
"... Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkari ..."
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Cited by 4 (1 self)
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Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Krz, Greene, and others), all of them based on the BorsukUlam theorem from algebraic topology, but otherwise quite di#erent. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. We show