Approximate Range Emptiness in Constant Time and Optimal Space

Abstract

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. *

Succinct Range Filters

“…The second theory-practice gap is that SuRF does not guarantee a theoretical false-positive rate for range queries based on the number of bits used, despite that it achieves good empirical results. Goswami et al [36] studied the theory aspect of the approximate range emptiness (i.e., range filtering) problem. They proved that any data structure that can answer approximate range emptiness queries has the worst-case space lower bound of Ω(n lg(L/ϵ )) − O (n) bits, where n represents the number of items, L denotes the maximum interval length for range queries (in SuRF, L equals to the size of the key space), and ϵ is the false-positive rate.…”

Near-Optimal Data Structure for Approximate Range Emptiness Problem in Information-Centric Internet of Things

“…Goswami et al [4] study the approximate range emptiness problem by focusing on the simple case in which S (the set to be represented) is static, rendering the derived lower bound and proposed data structure [4] unsuitable for the more general scenarios of IoT data streams, which are commonly seen in IoT environments [8], [14]- [17], [19]- [22].…”

“…Goswami et al [4] first define and study the approximate range emptiness problem, which generalizes the approximate membership query problem from querying a single point ''q ∈ S? to an interval ''[a, b] ∩ S = ∅?…”

“…As far as we know, it is currently the only work that is relevant to the approximate range emptiness problem. Both the space lower bound and an optimal data structure are proposed in [4].…”

“…Goswami et al [4] generalize the ε-approximate membership problem from asking whether a query element q (q ∈ U ) belongs to S or not to asking whether a query interval I = [a, b] of length L ([a, b] contains L integers) intersects with S or not, i.e., whether [a, b] ∩ S is ∅ or not. This generalized problem is called the approximate range emptiness problem, as studied in [4], and is defined as follows: the task of this problem is to represent a set of S of n elements selected from U by a memory-efficient data structure D, and then, for a query interval I of length L,…”