# Revisiting Graph Persistence for Updates and Efficiency

## TL;DR

This paper develops efficient algorithms for updating graph persistence, including ordinary and zigzag types, with significant improvements in update times for various cases, enhancing computational efficiency in topological data analysis.

## Contribution

It introduces novel algorithms for local updates in graph persistence, achieving logarithmic and sublinear update times, and improves existing algorithms for zigzag persistence.

## Key findings

- Switches in ordinary persistence can be done in O(log m) time.
- Zigzag persistence computation is improved to O(m log m) time.
- Various bar types in zigzag persistence can be updated efficiently in O(log m) or O(√m log m) time.

## Abstract

It is well known that ordinary persistence on graphs can be computed more efficiently than the general persistence. Recently, it has been shown that zigzag persistence on graphs also exhibits similar behavior. Motivated by these results, we revisit graph persistence and propose efficient algorithms especially for local updates on filtrations, similar to what is done in ordinary persistence for computing the vineyard. We show that, for a filtration of length $m$, (i) switches (transpositions) in ordinary graph persistence can be done in $O(\log m)$ time; (ii) zigzag persistence on graphs can be computed in $O(m\log m)$ time, which improves a recent $O(m\log^4n)$ time algorithm assuming $n$, the size of the union of all graphs in the filtration, satisfies $n\in\Omega({m^\varepsilon})$ for any fixed $0<\varepsilon<1$; (iii) open-closed, closed-open, and closed-closed bars in dimension $0$ for graph zigzag persistence can be updated in $O(\log m)$ time, whereas the open-open bars in dimension $0$ and closed-closed bars in dimension $1$ can be done in $O(\sqrt{m}\,\log m)$ time.