Optimal Hashing in External Memory

Abstract

Hash tables are a ubiquitous class of dictionary data structures. However, standard hash table implementations do not translate well into the external memory model, because they do not incorporate locality for insertions. Iacono and Patracsu established an update/query tradeoff curve for external hash tables: a hash table that performs insertions in $O(\lambda/B)$ amortized IOs requires $\Omega(\log_\lambda N)$ expected IOs for queries, where $N$ is the number of items that can be stored in the data structure, $B$ is the size of a memory transfer, $M$ is the size of memory, and $\lambda$ is a tuning parameter. They provide a hashing data structure that meets this curve for $\lambda$ that is $\Omega(\log\log M + \log_M N)$. Their data structure, which we call an \defn{IP hash table}, is complicated and, to the best of our knowledge, has not been implemented. In this paper, we present a new and much simpler optimal external memory hash table, the \defn{Bundle of Arrays Hash Table} (BOA). BOAs are based on size-tiered LSMs, a well-studied data structure, and are almost as easy to implement. The BOA is optimal for a narrower range of $\lambda$. However, the simplicity of BOAs allows them to be readily modified to achieve the following results: \begin{itemize} \item A new external memory data structure, the \defn{Bundle of Trees Hash Table} (BOT), that matches the performance of the IP hash table, while retaining some of the simplicity of the BOAs. \item The \defn{cache-oblivious Bundle of Trees Hash Table} (COBOT), the first cache-oblivious hash table. This data structure matches the optimality of BOTs and IP hash tables over the same range of $\lambda$. \end{itemize}