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An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology

Authors Tamal K. Dey, Tianqi Li, Yusu Wang

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Author Details

Tamal K. Dey
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA
Tianqi Li
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA
Yusu Wang
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA

Funding

This work is in part supported by National Science Foundation under grants CCF1740761, DMS-1547357, and RI-1815697.

Cite AsGet BibTex

Tamal K. Dey, Tianqi Li, and Yusu Wang. An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.36

Abstract

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their persistence. These concepts are well suited for spaces that are not directed. As a result, one needs a concept of homology that accommodates orientations in input space. Path-homology developed for directed graphs by Grigoryan, Lin, Muranov and Yau has been effectively adapted for this purpose recently by Chowdhury and Mémoli. They also give an algorithm to compute this path-homology. Our main contribution in this paper is an algorithm that computes this path-homology and its persistence more efficiently for the 1-dimensional (H₁) case. In developing such an algorithm, we discover various structures and their efficient computations that aid computing the 1-dimensional path-homology. We implement our algorithm and present some preliminary experimental results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • computational topology
  • directed graph
  • path homology
  • persistent path homology

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References

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