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.

References

Our sketch is based on the fully mergeable Q-digest sketch, but we do not know whether it is fully mergeable in its current form. We leave it as a future direction to come up with a fully mergeable mode for our algorithm.

— Optimal quantile estimation: beyond the comparison model  (2404.03847 - Gupta et al., 2024) in Practical considerations (Section 6), paragraph “Mergeability”

Indeed, it is not clear how to merge two elastic compactors so that the error can be bounded, as even for simpler relative compactors, the analysis under arbitrary merging is highly involved.

— Almost Optimal Relative Error Streaming Quantiles with Full Mergeability  (2610.18347 - Domes et al., 13 Oct 2026) in Section 2.2, paragraph 'Limitations'