A Space Lower Bound for Approximate Membership with Duplicate Insertions or Deletions of Nonelements

TL;DR

This paper establishes a fundamental space lower bound for approximate membership data structures that support insertions and deletions of elements, especially when duplicate insertions and deletions of non-elements are allowed, extending prior bounds.

Contribution

It proves a new space lower bound for approximate membership structures when the restrictions on duplicate insertions and deleting non-elements are lifted.

Findings

Lower bound of (1/2) * (1 - ε⁺ - 1/n) * log C(u, n) bits for general case

Shows the impact of lifting restrictions on space complexity

Extends understanding of space requirements for approximate membership data structures.

Abstract

Designs of data structures for approximate membership queries with false-positive errors that support both insertions and deletions stipulate the following two conditions: (1) Duplicate insertions are prohibited, i.e., it is prohibited to insert an element $x$ if $x$ is currently a member of the dataset. (2) Deletions of nonelements are prohibited, i.e., it is prohibited to delete $x$ if $x$ is not currently a member of the dataset. Under these conditions, the space required for the approximate representation of a datasets of cardinality $n$ with a false-positive probability of $\epsilon^{+}$ is at most $(1+o(1))n\cdot\log_2 (1/\epsilon^{+}) + O(n)$ bits [Bender et al., 2018; Bercea and Even, 2019]. We prove that if these conditions are lifted, then the space required for the approximate representation of datasets of cardinality $n$ from a universe of cardinality $u$ is at least $\frac 12 \cdot (1-\epsilon^{+} -\frac 1n)\cdot \log \binom{u}{n} -O(n)$ bits.