Cache-Oblivious Hashing

摘要

The hash table, especially its external memory version, is one of the most important index structures in large databases. Assuming a truly random hash function, it is known that in a standard external hash table with block size b, searching for a particular key only takes expected average t_q = 1 +1 /2~Ω(b) disk accesses for any load factor α bounded away from 1. However, such near-perfect performance is achieved only when b is known and the hash table is particularly tuned for working with such a blocking. In this paper we study if it is possible to build a cache-oblivious hash table that works well with any blocking. Such a hash table will automatically perform well across all levels of the memory hierarchy and does not need any hardware-specific tuning, an important feature in autonomous databases. We first show that linear probing, a classical collision resolution strategy for hash tables, can be easily made cache-oblivious but it only achieves t_q ＝ 1 +Θ(α/b) even if a truly random hash function is used. Then we demonstrate that the block probing algorithm (Pagh et al. in SIAM Rev. 53(3):547-558,2011) achieves t_q = 1 +1 /2~Ω(b), thus matching the cache-aware bound, if the following two conditions hold: (a) b is a power of 2; and (b) every block starts at a memory address divisible by b. Note that the two conditions hold on a real machine, although they are not stated in the cache-oblivious model. Interestingly, we also show that neither condition is dispensable: if either of them is removed, the best obtainable bound is t_q = 1 +O(α/b), which is exactly what linear probing achieves.