  In this paper, we study the role non-adaptivity plays in maintaining dynamic
data structures. Roughly speaking, a data structure is non-adaptive if the
memory locations it reads and/or writes when processing a query or update
depend only on the query or update and not on the contents of previously read
cells. We study such non-adaptive data structures in the cell probe model. This
model is one of the least restrictive lower bound models and in particular,
cell probe lower bounds apply to data structures developed in the popular
word-RAM model. Unfortunately, this generality comes at a high cost: the
highest lower bound proved for any data structure problem is only
polylogarithmic. Our main result is to demonstrate that one can in fact obtain
polynomial cell probe lower bounds for non-adaptive data structures.
  To shed more light on the seemingly inherent polylogarithmic lower bound
barrier, we study several different notions of non-adaptivity and identify key
properties that must be dealt with if we are to prove polynomial lower bounds
without restrictions on the data structures.
  Finally, our results also unveil an interesting connection between data
structures and depth-2 circuits. This allows us to translate conjectured hard
data structure problems into good candidates for high circuit lower bounds; in
particular, in the area of linear circuits for linear operators. Building on
lower bound proofs for data structures in slightly more restrictive models, we
also present a number of properties of linear operators which we believe are
worth investigating in the realm of circuit lower bounds.
