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21
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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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.
Zigzag Persistent Homology in Matrix Multiplication Time
 IN: PROCEEDINGS OF THE TWENTYSEVENTH ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY, 2011
, 2011
"... We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our al ..."
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Cited by 20 (0 self)
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We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our algorithm runs in O(M(n) + n 2 log 2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n 2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n 3) time in the worst case.
The tidy set: A minimal simplicial set for computing homology of clique complexes
 In Proc. ACM Symposium of Computational Geometry
, 2010
"... We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are pop ..."
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Cited by 17 (1 self)
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We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.
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.
Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps
"... We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A setvalued map of topdimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) c ..."
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Cited by 10 (8 self)
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We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A setvalued map of topdimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such setvalued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.
Cech Type Approach to Computing Homology of Maps, in preparation
"... Abstract. A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations ..."
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Cited by 8 (3 self)
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Abstract. A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of nedcessary input. We show that the approach is particularly advantegous in the case of homology map algorithms. 1. Introduction. Effective algorithms for computing homology of spaces and maps are needed in computer assisted proofs in dynamics based on topological tools (see [4, 17, 22, 23] and references therein). Recently, homology algorithms have also been used in robotics [30], material structure analysis [10, 11] and image recognition [3, 36],
Constructive MayerVietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes
, 2010
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Coreduction Homology Algorithm for Regular CWComplexes
, 2010
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An iterative algorithm for homology computation on simplicial shapes
, 2011
"... We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the comple ..."
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We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive MayerVietoris sequence, which relates the homology of a topological space to the homologies of its subspaces, i.e. the subcomponents of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the subcomponents and increases the efficiency of the algorithm.
Topological Data Analysis
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
, 2011
"... Scientific data is often in the form of a finite set of noisy points, sampled from an unknown space, and embedded in a highdimensional space. Topological data analysis focuses on recovering the topology of the sampled space. In this chapter, we look at methods for constructing combinatorial repres ..."
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Cited by 2 (0 self)
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Scientific data is often in the form of a finite set of noisy points, sampled from an unknown space, and embedded in a highdimensional space. Topological data analysis focuses on recovering the topology of the sampled space. In this chapter, we look at methods for constructing combinatorial representations of point sets, as well as theories and algorithms for effective computation of robust topological invariants. Throughout, we maintain a computational view by applying our techniques to a dataset representing the conformation space of a small molecule.