Approximate Range Emptiness in Constant Time and Optimal Space
Mayank Goswami, Allan Grønlund, Kasper Green Larsen, Rasmus Pagh · 2014
This paper studies the ε-approximate range emptiness problem, where the task is to represent a set S of n points from {0, …, U – 1} and answer emptiness queries of the form “[a; b] ∩ S ≠ ?” with a probability of false positives allowed. This generalizes the functionality of Bloom filters from single point queries to any interval length L. Setting the false positive rate to ε/L and performing L queries, Bloom filters yield a solution to this problem with space O(n lg(L/ε)) bits, false positive probability bounded by ε for intervals of length up to L, using query time O(L lg(L/ε)). Our first contribution is to show that the space/error trade-off cannot be improved asymptotically: Any data structure for answering approximate range emptiness queries on intervals of length up to L with false positive probability ε, must use space Ω(n lg(L/ε)) — O(n) bits. On the positive side we show that the query time can be improved greatly, to constant time, while matching our space lower bound up to a lower order additive term. This result is achieved through a succinct data structure for (non-approximate 1d) range emptiness/reporting queries, which may be of independent interest.