Adaptive dynamic data structures for the dynamic minimum problem

Determine the achievable per-operation access complexity of adaptive dynamic data structures that maintain a subset T ⊆ [m] and support insertions, deletions, and minimum-element queries when, for each updated element i, the data structure may adaptively choose which memory cells to read or write rather than being restricted to a fixed predetermined set S_i of cells.

Background

The paper considers a model where a data structure stores a set T ⊆ [m] in n memory cells (each holding an element of [m]) and supports add, delete, and minimum queries using at most t cell accesses per operation. In the non-adaptive setting, each update to element i is constrained to a fixed set S_i of cells. A binary search tree achieves t ≲ log m, and sunflower-based arguments imply t ≳ log m / log log m in the non-adaptive regime.

The open problem asks what efficiency is possible when adaptivity is allowed—i.e., when the set of accessed memory cells may depend on the contents read so far—removing the constraint of predetermined access sets S_i.