Vines and vineyards by updating persistence in linear time

D. Cohen Steiner, H. Edelsbrunner, D. Morozov, in:, ACM, 2006, pp. 119–126.

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Abstract
Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its per- tinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm’s anal- ysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.
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Date Published
2006-06-01
Acknowledgement
Partially supported by NSF under grant CCR- 00-86013, by DARPA under grant HR0011-05-1-0007, and by the Lawrence Livermore National Laboratory under grant B543154.
Page
119 - 126
Conference
SCG: Symposium on Computational Geometry
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Cohen Steiner D, Edelsbrunner H, Morozov D. Vines and vineyards by updating persistence in linear time. In: ACM; 2006:119-126. doi:10.1145/1137856.1137877
Cohen Steiner, D., Edelsbrunner, H., & Morozov, D. (2006). Vines and vineyards by updating persistence in linear time (pp. 119–126). Presented at the SCG: Symposium on Computational Geometry, ACM. https://doi.org/10.1145/1137856.1137877
Cohen Steiner, David, Herbert Edelsbrunner, and Dmitriy Morozov. “Vines and Vineyards by Updating Persistence in Linear Time,” 119–26. ACM, 2006. https://doi.org/10.1145/1137856.1137877.
D. Cohen Steiner, H. Edelsbrunner, and D. Morozov, “Vines and vineyards by updating persistence in linear time,” presented at the SCG: Symposium on Computational Geometry, 2006, pp. 119–126.
Cohen Steiner D, Edelsbrunner H, Morozov D. 2006. Vines and vineyards by updating persistence in linear time. SCG: Symposium on Computational Geometry 119–126.
Cohen Steiner, David, et al. Vines and Vineyards by Updating Persistence in Linear Time. ACM, 2006, pp. 119–26, doi:10.1145/1137856.1137877.

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