Lower Bounds for Semi-adaptive Data Structures via Corruption

In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of {\em semi-adaptive\} data structures, which compute the Disjointness function. There, the data are subsets $x_1,\dots,x_k$ and $y$ of $\{1,\dots,n\}$, the updates can modify $y$ (by inserting and removing elements), and the queries are an index $i \in \{1,\dots,k\}$ (query $i$ should answer whether $x_i$ and $y$ are disjoint, i.e., it should compute the Disjointness function applied to $(x_i, y)$). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.