{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/elder-rule-staircodes-for-augmented-metric","title":"Elder-Rule-Staircodes for Augmented Metric Spaces","arxiv_id":"2003.04523","date":"2020-03-10","proceeding":null,"authors":["Chen Cai","Woojin Kim","Facundo Memoli","Yusu Wang"],"abstract":"An augmented metric space is a metric space $(X, d_X)$ equipped with a function $f_X: X \\to \\mathbb{R}$. This type of data arises commonly in practice, e.g, a point cloud $X$ in $\\mathbb{R}^d$ where each point $x\\in X$ has a density function value $f_X(x)$ associated to it. An augmented metric space $(X, d_X, f_X)$ naturally gives rise to a 2-parameter filtration $\\mathcal{K}$. However, the resulting 2-parameter persistent homology $\\mathrm{H}_{\\bullet}(\\mathcal{K})$ 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 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode $\\mathrm{H}_0(\\mathcal{K})$. Specifically, if $n = |X|$, the elder-rule-staircode consists of $n$ number of staircase-like blocks in the plane. We show that if $\\mathrm{H}_0(\\mathcal{K})$ is interval decomposable, then the barcode of $\\mathrm{H}_0(\\mathcal{K})$ is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of $\\mathrm{H}_0(\\mathcal{K})$ can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in $O(n^2\\log n)$ time, which can be improved to $O(n^2\\alpha(n))$ if $X$ is from a fixed dimensional Euclidean space $\\mathbb{R}^d$, where $\\alpha(n)$ is the inverse Ackermann function.","url_abs":"http://arxiv.org/abs/2003.04523v2","url_pdf":"http://arxiv.org/pdf/2003.04523v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"elder-rule-staircodes-for-augmented-metric","repo_url":"https://github.com/Chen-Cai-OSU/ER-staircode","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}