Full mergeability of the O(1/ε)-word quantile sketch

Determine whether the deterministic O(1/ε)-word bounded-universe quantile sketch introduced in this paper is fully mergeable—that is, whether two sketches with the same error parameter ε can be merged into a single sketch that preserves the same ε guarantee without any additional error; if not, develop a fully mergeable mode achieving the same asymptotic space and error bounds.

Background

Mergeability is a desired property of quantile sketches. The q-digest is fully mergeable, while GK and optimal KLL are not (KLL has a fully mergeable mode that loses its optimal space). The new sketch builds on ideas from q-digest but its full mergeability is unknown.

The authors also describe a partial-merge procedure that adds only O(t / log U) discrepancy when merging two instances of sizes t and t′, indicating some progress but leaving open whether exact full mergeability with no error increase can be achieved.