Near-optimal hash count with inverse-slack operation time

Establish whether \(O(\epsilon^{-1})\)-time operations are achievable for d-ary cuckoo hashing when \(d=\ln \epsilon^{-1}+o(1)\).

Background

The main result requires an additive positive constant α\alpha in the hash count, namely d≥ln⁡ϵ−1+αd\geq \ln\epsilon^{-1}+\alpha. The unresolved question is whether this constant additive slack can be reduced to an asymptotically vanishing term while retaining operation time proportional to ϵ−1\epsilon^{-1}.

References

Can one hope for $O(\epsilon{-1})$-time operations even when $d = \ln \epsilon{-1} + o(1)$? Our bounds require $d \ge \ln \epsilon{-1} + \alpha$ for some small but positive constant $\alpha$.

— Efficient $d$-ary Cuckoo Hashing at High Load Factors by Bubbling Up  (2501.02312 - Kuszmaul et al., 4 Jan 2025) in Section Conclusion