Fast local searches and updates in bounded universes

Given a bounded universe {0, 1, . . . , U−1}, we show how to perform (successor) searches in O(log log ∆) expected time and updates in O(log log ∆) expected amortized time, where ∆ is the rank difference between the element being searched for and its successor in the structure. This unifies the results of traditional bounded universe structures (which support successor searches in O(log logU) time) and hashing (which supports exact searches in O(1) time). We also show how these results can be extended to answer approximate nearest neighbour queries in low dimensions.