Breaking the Quadratic Barrier: Robust Cardinality Sketches for Adaptive Queries

TL;DR

This paper introduces robust cardinality sketches that can handle exponentially many adaptive queries by limiting the number of queries per element, overcoming previous quadratic barriers in query adaptivity.

Contribution

We develop new estimators that withstand exponential adaptive queries by controlling per-element query participation, advancing the robustness of cardinality sketches.

Findings

Handles exponential adaptive queries with per-element query limits

Shifts quadratic barrier from total queries to per-element queries

Expands robust algorithm design toolkit

Abstract

Cardinality sketches are compact data structures that efficiently estimate the number of distinct elements across multiple queries while minimizing storage, communication, and computational costs. However, recent research has shown that these sketches can fail under {\em adaptively chosen queries}, breaking down after approximately $\tilde{O}(k^2)$ queries, where $k$ is the sketch size. In this work, we overcome this \emph{quadratic barrier} by designing robust estimators with fine-grained guarantees. Specifically, our constructions can handle an {\em exponential number of adaptive queries}, provided that each element participates in at most $\tilde{O}(k^2)$ queries. This effectively shifts the quadratic barrier from the total number of queries to the number of queries {\em sharing the same element}, which can be significantly smaller. Beyond cardinality sketches, our approach expands the toolkit for robust algorithm design.