Cycle bases of graphs and sampled manifolds

Gotsman, Craig; Kaligosi, Kanela; Mehlhorn, Kurt; Michail, Dimitrios; Pyrga, Evangelia

doi:10.1016/j.cagd.2006.07.001

Cited by 7 publications

(2 citation statements)

References 34 publications

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“…The algorithm presented in [11] extracts two types of possible 1-cycles which identify handles and tunnels on 2-manifold surfaces. In [23], another method has been proposed for computing the non-contractible 1-cycles on smooth compact 2-manifolds. The shape of the computed generators has been addressed in [39,4].…”

Section: Related Workmentioning

confidence: 99%

An iterative algorithm for homology computation on simplicial shapes

Boltcheva¹,

Canino²,

Aceituno³

et al. 2011

Computer-Aided Design

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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 sub-components. 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 Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components 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 sub-components and increases the efficiency of the algorithm.

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Section: Related Workmentioning

confidence: 99%

An iterative algorithm for homology computation on simplicial shapes

Boltcheva¹,

Canino²,

Aceituno³

et al. 2011

Computer-Aided Design

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“…As an application of the APSP problem, our second problem (denoted by MCB) is to obtain a minimum weight cycle basis of a weighted graphessentially, to find a set of basis cycles with the least total weight such that every other cycle can be represented as a linear combination of the basis cycles. The MCB problem has applications to problems in biochemistry [14], three-dimensional surface reconstruction from a point cloud [15] and electric networks [11].…”

Section: Introductionmentioning

confidence: 99%

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

Dutta

Chaitanya

Kothapalli

et al. 2018

IJNC

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Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths (APSP) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi-and manycore architectures.It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures.In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortest-paths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.

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Building a Weighted Complex from Data

Grady

Polimeni

2010

Discrete Calculus

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Cycle bases of graphs and sampled manifolds

Cited by 7 publications

References 34 publications

An iterative algorithm for homology computation on simplicial shapes

An iterative algorithm for homology computation on simplicial shapes

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

Building a Weighted Complex from Data

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