Computing persistence over changing filtrations give rise to a stack of 2D persistence diagrams where the birth-death points are connected by the so-called `vines'. We consider computing these vines over changing filtrations for zigzag persistence. We observe that eight atomic operations are sufficient for changing one zigzag filtration to another and provide update algorithms for each of them. Six of these operations that have some analogues to one or multiple transpositions in the non-zigzag case can be executed as efficiently as their non-zigzag counterparts. This approach takes advantage of a recently discovered algorithm for computing zigzag barcodes by converting a zigzag filtration to a non-zigzag one and then connecting barcodes of the two with a bijection. The remaining two atomic operations do not have a strict analogue in the non-zigzag case. For them, we propose algorithms based on explicit maintenance of representatives (homology cycles) which can be useful in their own rights for applications requiring explicit updates of representatives.
翻译:计算机在改变过滤过程中的持久性导致产生一系列2D持久性图,其中出生-死亡点与所谓的“维奈斯”相连。我们考虑将这些藤用于改变过滤以保持zigzag持久性。我们发现,八种原子操作足以将一个zigzag过滤器转换成另一个zigzag过滤器,并为每个原子提供更新算法。这些操作中有6种在非zigzag案中具有某种类似或多种转换器的模拟作用的操作可以像其非zigzag对应方一样高效地执行。这个方法利用最近发现的计算zigzag条形码的算法,将zigzag过滤器转换为非zigzag过滤器,然后将2个条形编码与两条形相连接。其余两种原子操作在非zigzag案中没有严格的类比。对于它们来说,我们建议基于明确维护代表(分子周期)的算法,这种算法在他们自己的应用中有用,要求代表明确更新。