Dynamic Linear Product queries: optimality and lower bounds

Determine whether a linear-time preprocessing algorithm that supports dynamic Linear Product operations on a sequence over an associative semigroup—specifically, updating the value of an element, inserting a new element, and deleting an element—while answering range product queries s_i ⊗ s_{i+1} ⊗ ⋯ ⊗ s_j in O(log n) time per operation is optimal; moreover, derive nontrivial lower bounds and characterize the trade-off between preprocessing time and processing (query/update) time in this dynamic setting.

Background

The paper establishes tight preprocessing-query time trade-offs for static Linear Product queries over semigroups, achieving optimal bounds under stated assumptions. However, all results apply to static inputs.

In the dynamic Linear Product setting, the authors identify three operations (update an element, insert an element, delete an element) and note that a simple linear preprocessing approach yields O(log n) time per operation and query. They explicitly state that the optimality of this bound is unknown and that no nontrivial lower bounds or trade-offs have been proved for the dynamic case.