The union of balls and its dual shape
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
An incremental algorithm for Betti numbers of simplicial complexes
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Computational geometry: a retrospective
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Computing Betti numbers via combinatorial Laplacians
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Algorithms for manifolds and simplicial complexes in Euclidean 3-space (preliminary version)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Homology algorithm based on acyclic subspace
Computers & Mathematics with Applications
Coreduction homology algorithm for inclusions and persistent homology
Computers & Mathematics with Applications
A new approach to crushing 3-manifold triangulations
Proceedings of the twenty-ninth annual symposium on Computational geometry
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An algorithm for computing the homology type of a triangulation is analyzed. By triangulation is meant a finite simplicial complex; its homology type is given by its homology groups (with integer coefficients). The algorithm could be used in computer-aided design to tell whether two finite-element meshes or Bezier-spline surfaces are of the same topological type, and whether they can be embedded in R/sup 3/. Homology computation is a pure combinatorial problem of considerable intrinsic interest. While the worst-case bounds obtained for this algorithm are poor, it is argued that many triangulations (in general) and virtually all triangulations in design are very sparse in a particular sense. This sparseness measure is formalized, and a probabilistic analysis of the sparse case is performed to show that the expected running time, of the algorithm is roughly quadratic in the geometric complexity (number of simplices) and linear in the dimension.