Recommendable!
After trial and error, they discovered and confirmed that the theoretical lower bound was the same as the previously established upper bound. Case closed! đ
"... Hash tables are among the oldest, simplest, fastest and most widely used data structures today. They’re designed to perform three basic operations: insertions, which add new items to the database; queries, which access an item or check to see whether it exists; and deletions. ..."
From the abstract:
"For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Theta(log n) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglog n) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(loglog n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters).
This paper shows that O(loglog n) wasted bits per key is not the end of the line for hashing. In fact, for any k \in [log* n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(\log^{(k)} n) wasted bits per key (all with high probability in n). This means that, each time we increase insertion/deletion time by an \emph{additive constant}, we reduce the wasted bits per key \emph{exponentially}. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct."
On the Optimal Time/Space Tradeoff for Hash Tables (open access)
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