Bucketized cuckoo hashing is a practically efficient hash table scheme in which each object $u$ is stored in one of two buckets $h_1(u), h_2(u)$ of capacity $\ell$. For any bucket size $\ell\in\mb{N}$, there is a threshold $ε^*(\ell)=(2/e)^\ell\mathrm{poly}(\ell)$ for which there exists a way to fill the hash table to any load factor less than $1-ε^*$ with low probability of an error. Queries and deletions only need to check two buckets to find whether an object exists. Our contribution is to give a new insertion procedure for bucketized cuckoo hashing. For any $δ\in[.99^\ell,1]$, our algorithm can fill the hash table to load factor $1-ε=1-(1+δ)(ε^*)$ with an expected run time of $O(δ^{-1}(ε^*)^{-1})$ per insertion. This gives the first $\mathrm{poly}(ε^{-1})$ insertion time bound, and the first $f(ε^{-1})$ time bound for load factors that are very close to the optimal threshold. Additionally, our algorithm (which can be viewed as a variation of the classic random-walk algorithm) comes with a very strong amortized guarantee: it performs $O(1)$ amortized expected evictions per insertion. Furthermore, we show that the traditional random-walk algorithm cannot match this guarantee. Finally, our insertion protocol also comes with the feature that, for any key $u$ in the hash table, the query algorithm can \emph{guess} which of the two bins $h_1(u), h_2(u)$ the key $u$ is in with probability $1 - o(1)$ of being correct. Thus positive queries can complete in $1 + o(1)$ expected bin accesses.

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