Computing Zigzag Vineyard Efficiently Including Expansions and Contractions

TL;DR

This paper extends the FastZigzag algorithm to efficiently handle expansions and contractions in zigzag persistence, achieving quadratic time complexity comparable to non-zigzag cases.

Contribution

It demonstrates that expansions and contractions can be performed in quadratic time within the zigzag persistence framework, matching non-zigzag complexities.

Findings

Expansions and contractions can be implemented in quadratic time in zigzag persistence.

The half-way constructed up-down filtration enables linear and quadratic time operations.

The approach matches the efficiency of non-zigzag algorithms for these operations.

Abstract

Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm, expansions and contractions pose difficulty in breaking the barrier of cubic complexity. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case.