Elder-Rule-Staircodes for Augmented Metric Spaces

Authors Chen Cai, Woojin Kim, Facundo Mémoli, Yusu Wang

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Author Details

Chen Cai
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Woojin Kim
  • Department of Mathematics, The Ohio State University, Columbus, OH, USA
Facundo Mémoli
  • Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Yusu Wang
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA


The authors thank to the anonymous reviewers who made a number of helpful comments to improve the paper. Also, CC and WK thank Cheng Xin for helpful discussions.

Cite AsGet BibTex

Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang. Elder-Rule-Staircodes for Augmented Metric Spaces. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Topology
  • Theory of computation → Computational geometry
  • Persistent homology
  • Multiparameter persistence
  • Barcodes
  • Elder rule
  • Hierarchical clustering
  • Graded Betti numbers


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