Bounds for linear and uniform probing with deletions in open addressing without reordering

Determine whether linear probing and uniform probing (without reordering) achieve expected insertion times, probe complexities, and amortized probe complexities that are bounded as a function of δ^{-1} in the open-addressing setting that supports both insertions and deletions over an infinite time horizon.

Background

The paper focuses on open-addressing hash tables without reordering in the insertion-only setting, where it provides optimal bounds. It then highlights that when the model is extended to allow both insertions and deletions over an infinite time horizon (still without reordering), even basic schemes such as linear probing and uniform probing remain poorly understood.

Specifically, the authors note that it is not known whether these basic schemes achieve expected insertion times, probe complexities, or amortized probe complexities that are bounded functions of the inverse slack δ^{-1}. They also point out that, in this dynamic setting, the optimal amortized expected probe complexity is δ^{-Ω(1)}, implying that constant or logarithmic-in-δ bounds like those obtained in the insertion-only setting are impossible, but leaving open whether linear or uniform probing meet any δ-bounded guarantees.