Approximate Range Emptiness in Constant Time and Optimal Space

This paper studies the varepsilon-approximate range emptiness problem, where the task is to represent a set S of n points from 0,ldots,U-1 and answer emptiness queries of the form "[a ; b]cap S neq emptyset ?" 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 varepsilon/L and performing L queries, Bloom filters yield a solution to this problem with space O(n lg(L/varepsilon)) bits, false positive probability bounded by varepsilon for intervals of length up to L, using query time O(L lg(L/varepsilon)). 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 varepsilon, must use space Omega(n lg(L/varepsilon)) - 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.