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

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 $ε^{+}$ is at most $(1+o(1))n\cdot\log_2 (1/ε^{+}) + O(n)$ 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 $\frac 12 \cdot (1-ε^{+} -\frac 1n)\cdot \log \binom{u}{n} -O(n)$ bits.