Fast Insertion for Bucketized Cuckoo Hashing

2026-07-27

AI Summary

This work addresses the inefficiency of insertions in bucketed cuckoo hashing under high load factors by proposing a novel insertion algorithm equipped with a prediction mechanism. By employing an improved random walk strategy, the method achieves—for the first time—a polynomial expected insertion time of $O(\delta^{-1}(\varepsilon^*)^{-1})$ with respect to $(\varepsilon^*)^{-1}$ when the load factor approaches the theoretical optimum $1 - (1+\delta)\varepsilon^*$, where $\delta \in [0.99^\ell, 1]$. It also guarantees $O(1)$ amortized expected eviction cost. Furthermore, during queries, the algorithm predicts the correct bucket for an element with probability $1 - o(1)$, enabling successful lookups with only $1 + o(1)$ bucket accesses and thereby significantly enhancing query performance.

Abstract

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.

Problem

• bucketized cuckoo hashing
• insertion time
• load factor
• hash table
• amortized evictions

Innovation

• bucketized cuckoo hashing
• fast insertion
• amortized eviction
• load factor threshold
• probabilistic query optimization