Dynamic perfect hashing: upper and lower bounds

The dynamic dictionary problem is considered: provide an algorithm for storing a dynamic set, allowing the operations insert, delete, and lookup. A dynamic perfect hashing strategy is given: a randomized algorithm for the dynamic dictionary problem that takes $O(1)$ worst-case time for lookups and $O(1)$ amortized expected time for insertions and deletions; it uses space proportional to the size of the set stored. Furthermore, lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved. This class encompasses realistic hashing-based schemes that use linear space. Such algorithms have amortized worst-case time complexity $\Omega (\log n)$ for a sequence of n insertions and lookups; if the worst-case lookup time is restricted to $k$, then the lower bound becomes $\Omega (k \cdot n^{{1 / k}} )$.