Constant-bucket-size insertion near the critical load threshold

Determine whether bucketized cuckoo hashing with constant bucket size can approach arbitrarily closely the critical load threshold while maintaining \(O(1)\) expected insertion time.

Background

The critical load threshold is the maximum load for which a valid hash-table configuration exists. The paper leaves open whether constant-size buckets permit operation arbitrarily close to this existential threshold without sacrificing constant expected insertion time.

References

If the bucket size $b$ is a constant, can one get arbitrarily close to the critical load threshold (the maximum load at which a valid hash-table configuration exists) while still supporting $O(1)$-expected time insertions?

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