Updating zigzag representatives efficiently

2026-07-17

AI Summary

This work addresses the inefficiency in updating persistence representatives under dynamic zigzag filtrations, where changes such as insertion or deletion of simplices alter adjacency relations and hinder computational performance. To overcome this challenge, the paper introduces a novel algorithm for extracting and updating zigzag persistence representatives based on the R = DV decomposition commonly used in non-zigzag settings. This approach achieves, for the first time, efficient maintenance of zigzag representatives amid evolving adjacency structures, employing an update strategy with quadratic time complexity. The proposed method significantly narrows the computational gap between zigzag and non-zigzag persistence computations, thereby substantially enhancing the efficiency of persistent homology calculations over dynamic filtrations.

Abstract

Computation of zigzag persistence has progressed in recent years, with results showing that complexities of many problems closely align with those in the non-zigzag setting. The major efficiency gap now lies in the updating of zigzag representatives. In this paper, we propose efficient algorithms for updating zigzag representatives based on a recent algorithm for extracting zigzag representatives from a $R=DV$ decomposition of a constructed non-zigzag. The main difficulty for designing our update algorithms lies in the adjacency change occurring in two operations that elongate or shorten a filtration. Despite the adjacency change, we find that the update can still be done efficiently in quadratic time.

Problem

• zigzag persistence
• representatives update
• filtration
• computational topology
• efficiency

Innovation

• zigzag persistence
• representative update
• R=DV decomposition
• filtration update
• quadratic time algorithm