Obtain polynomial insertion time with logarithmic bucket size

Establish whether bucketized cuckoo hashing with bucket size b=Theta(log epsilon^{-1}) can support polynomial-in-epsilon^{-1} expected-time insertions while operating at load factor 1-epsilon.

Background

The paper identifies bucketized cuckoo hashing as a harder setting than ordinary d-ary cuckoo hashing because increasing bucket size does not proportionally increase the available independent randomness. It specifically asks whether polynomial insertion time is possible when the bucket size grows logarithmically with epsilon{-1}.

References

In addition to these, another major open question is to develop efficient insertion algorithms for bucketized cuckoo hashing, where each element hashes to two buckets of size $b$ . This setting appears to be harder to analyze than the $d$-ary case because, as we increase $b$, the total amount of randomness that we have to work with does not increase. Because each item hashes to only two buckets, any item that is evicted more than once in its lifetime will be forced to make use of spoiled randomness (randomness that has already affected the hash-table state in the past). This issue of spoiled randomness seems to be a major challenge for the analysis of the bucketized version of the data structure. Within the study of bucketized cuckoo hashing, there are several goals that would be interesting to accomplish. (See, also, the discussion of bucketized cuckoo hashing in the related-work portion of the introduction.) Major questions include: Can one achieve $poly \epsilon{-1}$-time insertions with buckets of size $b = \Theta(\log \epsilon{-1})$?

— Efficient $d$-ary Cuckoo Hashing at High Load Factors by Bubbling Up  (2501.02312 - Kuszmaul et al., 4 Jan 2025) in Conclusion, paragraph beginning “Within the study of bucketized cuckoo hashing”