3.5-Way Cuckoo Hashing for the Price of 2-and-a-Bit

Abstract

Abstract. The study of hashing is closely related to the analysis of balls and bins; items are hashed to memory locations much as balls are thrown into bins. In particular, Azar et. al.[2] considered putting each ball in the less-full of two random bins. This lowers the probability that a bin exceeds a certain load from exponentially small to doubly exponential, giving maximum load log log n + O(1) with high probability. Cuckoo hashing [20] draws on this idea. Each item is hashed to two buckets of capacity k. If both are full, then the insertion procedure moves previously-inserted items to their alternate buckets to make space for the new item. In a natural implementation, the buckets are represented by partitioning a fixed array of memory into non-overlapping blocks of size k. An item is hashed to two such blocks and may be stored at any location within either one. We analyze a simple twist in which each item is hashed to two arbitrary size-k memory blocks. (So consecutive blocks are no longer disjoint, but rather overlap by k − 1 locations.) This twist increases the space utilization from 1 − (2/e + o(1)) k to 1 − (1/e + o(1)) 1.59k in general. For k = 2, the new method improves utilization from 89.7% to 96.5%, yet lookups access only two items at each of two random locations. This result is surprising because the opposite happens in the non-cuckoo setting; if items are not moved during later insertions, then shifting from non-overlapping to overlapping blocks makes the distribution less uniform.

A Cuckoo Hashing Variant with Improved Memory Utilization and Insertion Time

“…Ref. [2] do not provide a theoretical memory utilization threshold for small k. The empirical results of Ref. [2] show that in the CUCKOO-OVERLAP algorithm β Overlap (k = 2, d = 2, t = n) = 0.9650 and β Overlap (k = 3, d = 2, t = n) = 0.9945.…”

“…Empirical results show that our modification dramatically decreases the number of candidate look-ups required to insert an item compared to Ref. [2]. In the overlapping buckets variant [2], some buckets are split between two pages, so that each item resides in up to 2d pages.…”

“…[2]. In the overlapping buckets variant [2], some buckets are split between two pages, so that each item resides in up to 2d pages. In our variant, each bucket is fully contained in a single page.…”

Wear Minimization for Cuckoo Hashing: How Not to Throw a Lot of Eggs into One Basket

“…Several variants of cuckoo hashing have been considered, including the use of a small cache (called a "stash") and the use of more than two hash functions (e.g., see [2,11,12,14,20]). Also, as mentioned above, there has also been work improving the space complexity [13,22], as well as experimental analyses of cuckoo hashing [1,35]. Nevertheless, we are not familiar with any previous work on the memory-wear performance of cuckoo hashing, from either a theoretical or experimental viewpoint.…”

“…For instance, the cuckoo hashing with a stash variation sacrifices some of the elegance of the basic algorithm by adding an auxiliary cache to break insertion cycles, greatly reducing the failure probability of the algorithm and reducing the need to rebuild the structure when a failure happens [20]. There have also been improvements to the space complexity for cuckoo hashing [13,22], as well as experimental studies of the practical applications of cuckoo hashing [1,35]. In spite of its elegance, efficiency, and practicality, however, cuckoo hashing has a major drawback with respect to modern memory technologies.…”