In this work, we continue the examination of the role non-adaptivity} plays in maintaining dynamic data structures, initiated by Brody and Larsen [BL15].. We consider nonadaptive data structures for predecessor search in the w-bit cell probe model. Predecessor search is one of the most well-studied data structure problems. For this problem, using non-adaptivity comes at a steep price. We provide exponential cell probe complexity separations between (i) adaptive and non-adaptive data structures and (ii) non-adaptive and memoryless data structures for predecessor search.

A classic data structure of van Emde Boas [vEB75] solves dynamic predecessor search in O ( log log m ) probes; this data structure is adaptive. For dynamic data structures which make nonadaptive updates, we show the cell probe complexity is O min log m log ( w log m ) w n log m . We also give a nearly-matching min log w log m w log w n log m lower bound. We also give an ( m ) lower bound for memoryless data structures. Our lower bound technique is tailored to nonadaptive (as opposed to memoryless) updates and should be of independent interest.