Random-walk and BFS performance for bucketized cuckoo hashing

Characterize whether random-walk and breadth-first-search algorithms for bucketized cuckoo hashing achieve either polynomial-in-\(\epsilon^{-1}\) insertion time with \(b=\Theta(\log\epsilon^{-1})\) or arbitrarily near-threshold load at constant bucket size.

Background

The paper specifically identifies the random-walk and breadth-first-search policies as unresolved in the bucketized setting. The question concerns whether either policy can attain one of the two preceding goals: polynomial insertion time with logarithmic bucket size, or constant expected insertion time arbitrarily close to the critical threshold with constant bucket size.