3 research outputs found
Fast Computation of Zigzag Persistence
Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a ?-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares
Computing Zigzag Vineyard Efficiently Including Expansions and Contractions
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
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum