A data structure for quotient flag complexes

2026-09-26

AI Summary

This study addresses the efficiency and structural bottlenecks arising from the inability of simplex trees in flag complexes to implicitly store attaching data, and proposes the QF-tree data structure. By labeling collapsed faces and indexing quotient vertex words, this method efficiently encodes the cells and attaching maps of quotient CW-complexes. Grounded in a theoretical proof that finite dimensions determine the entire structure, it achieves linear space complexity and local updates, significantly outperforming homotopy-equivalent cone models. Furthermore, by integrating techniques such as zigzag persistence and Trie prefix trees, this work decouples quotient maintenance from algebraic computation costs. A comprehensive function library supporting complex topological operations is also provided, validating the compact and efficient computational advantages of the proposed approach.

Abstract

Vietoris-Rips filtrations, which are standard in topological data analysis, are flag complexes, and a simplex tree stores these without any attaching data. In this paper we ask what survives of this economy when a flag complex $K$ is divided by a subcomplex $A$, each connected component of $A$ being crushed to a point. Such a quotient is a CW complex whose cells are the simplices of $K\setminus A$, but their attaching maps are no longer implicit. We show that for flag $K$ the face order of the quotient is strictly graded exactly when $A$ is flag, and that the surviving labelled cells are determined by those of dimension at most 3. For $m$-flag pairs the threshold is $2m+1$, and it drops to $m+2$ when $K$ is flag. The prescribed cells form a regular CW decomposition only when $A$ is full in $K$. These results justify the QF-tree: a cell table that stores, for each surviving simplex, its ordered list of $d+1$ facets with collapsed facets flagged, indexed by a trie of quotient-vertex words. For bounded dimension its size is linear in the number of surviving simplices plus the retained provenance, and we derive and verify a simple formula for the collapsed fraction above which it is smaller than the homotopy-equivalent cone model. Because a collapse changes the attaching data only on the closed star of $A$, the QF-tree can also be applied locally inside a simplex tree. For a ball-shaped $A$ in the sampled Vietoris--Rips regime the closed star is a thin shell, and the median compact budget is below the cone model at every sampled radius. An accompanying library, modelled on Gudhi, implements the QF-tree, its local variant, an editable layer with local quotient updates, gluing, disc attachment, induced maps, cup products, fundamental-group presentations and zigzag persistence, and provided experiments separate the cost of maintaining a quotient from the cost of the algebra computed on it.

Problem

• quotient flag complex
• topological data analysis
• Vietoris-Rips filtration
• CW complex
• data structure

Innovation

• quotient flag complex
• QF-tree
• Vietoris-Rips filtration
• CW complex
• topological data analysis