We present a strongly history independent (SHI) hash table that supports search in O(1) worst-case time, and insert and delete in O(1) expected time using O(n) data space. This matches the bounds for dynamic perfect hashing, and improves on the best previous results by Naor and Teague on history independent hashing, which were either weakly history independent, or only supported insertion and search (no delete) each in O(1) expected time.

The results can be used to construct many other SHI data structures. We show straightforward constructions for SHI ordered dictionaries: for n keys from {1, . . . , n^k} searches take O(log log n) worst-case time and updates (insertions and deletions) O(log log n) expected time, and for keys in the comparison model searches take O(log n) worst-case time and updates O(log n) expected time. We also describe a SHI data structure for the order-maintenance problem. It supports comparisons in O(1) worst-case time, and updates in O(1) expected time. All structures use O(n) data space.