Dynamic Tree Product queries: optimality

Establish whether a linear-time preprocessing algorithm that supports dynamic Tree Product operations on a tree over an associative semigroup—specifically, updating the value at a vertex, linking two trees by adding an edge, and cutting a tree by deleting an edge—while answering path product queries between any two vertices in O(log n) time per operation is optimal.

Background

For the static Tree Product problem, the paper gives preprocessing schemes with optimal trade-offs and efficient parallelizations. The dynamic version considers updates to vertex values as well as structural changes via link and cut operations.

Using ideas from known dynamic tree data structures, the authors note that linear preprocessing can support all dynamic operations and O(log n)-time queries, but they explicitly state that it is unknown whether this bound is optimal.