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. 1

Citations