  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.
