Tight Bounds and Phase Transitions for Incremental and Dynamic Retrieval

Retrieval data structures are data structures that answer key-value queries without paying the space overhead of explicitly storing keys. The problem can be formulated in four settings (static, value-dynamic, incremental, or dynamic), each of which offers different levels of dynamism to the user. In this paper, we establish optimal bounds for the final two settings (incremental and dynamic) in the case of a polynomial universe. Our results complete a line of work that has spanned more than two decades, and also come with a surprise: the incremental setting, which has long been viewed as essentially equivalent to the dynamic one, actually has a phase transition, in which, as the value size $v$ approaches $\log n$, the optimal space redundancy actually begins to shrink, going from roughly $n \log \log n$ (which has long been thought to be optimal) all the way down to $Θ(n)$ (which is the optimal bound even for the seemingly much-easier value-dynamic setting).