Improve deterministic comparison-based relative-error quantile space

Determine whether the dependence on the failure probability can be improved for randomized comparison-based relative-error quantile sketches, thereby yielding a deterministic comparison-based algorithm with space below the current bound and closing the known logarithmic gap between deterministic lower and upper bounds.

Background

The paper derives a deterministic comparison-based algorithm from its randomized sketch by choosing an extremely small failure probability. This gives a space bound of O(ε^{-1} log^3(εN)), matching the current state of the art for deterministic comparison-based algorithms.

The authors observe that improving the dependence of the randomized algorithm's space on δ would also improve deterministic algorithms. Existing deterministic comparison-based lower and upper bounds differ by a factor of log(εN), and the paper explicitly leaves open whether this gap can be reduced. The question is relevant to both existential and constructive derandomization approaches.