Cache-Oblivious Iterated Predecessor Queries via Range Coalescing

Abstract

In this paper we develop an optimal cache-oblivious data structure that solves the iterated predecessor problem. Given k static sorted lists $L_1, L_2, \ldots , L_k$ of average length n and a query value q, the iterated predecessor problem is to find the largest element in each list which is less than q. Our solution to this problem, called “range coalescing”, requires $O(\log _{B+1} n + k/B)$ memory transfers for a query on a cache of block size B, which is information-theoretically optimal. The range-coalescing data structure consumes O(k n) space, and preprocessing requires only O(kn / B) memory transfers with high probability, given a tall cache of size $M = \varOmega \mathopen {}\left( B^2\right) \mathclose {}$.