Redundancy–universe-size trade-offs for smaller universes

Determine the precise trade-offs between universe size and redundancy for dynamic ordered dictionaries in the standard Word-RAM model when the universe satisfies \(\log u=o(w)\), where \(u\) is the universe size and \(w\) is the word size.

Background

The paper establishes optimal redundancy–time bounds for dynamic rank/select dictionaries in the regime log⁡u=Θ(w)\log u=\Theta(w), while also obtaining improved results for log⁡u=O(w)\log u=O(\sqrt w) in the standard Word-RAM model and for log⁡u=w/polylog⁡w\log u=w/\operatorname{polylog} w in the cell-probe model. These results indicate a sharp distinction between large and smaller universes.

The unresolved issue is to characterize the full dependence of achievable redundancy on the universe size throughout the intermediate regime log⁡u=o(w)\log u=o(w).

References

We ask what exactly are the trade-offs of universe size to redundancy when $\log u=o(w)$.

— Achieving Optimal Redundancy for Small Dynamic Rank/Select Dictionaries  (2610.01853 - Domingues, 1 Oct 2026) in Section “Discussion and Open problems,” paragraph “Smaller universes”

Further work is required to achieve worst-case times for $o(n)$ redundancy.

— Achieving Optimal Redundancy for Small Dynamic Rank/Select Dictionaries  (2610.01853 - Domingues, 1 Oct 2026) in Section “Discussion and Open problems,” paragraph “Achieving $o(n)$ redundancy”