Subquadratic extraction of smaller emulators

Construct an algorithm that, given the second-layer emulator J for the terminal set Y, extracts a smaller reachability emulator for the relevant planar digraph and terminal set in subquadratic time, thereby enabling additional layers in the offline dynamic reachability scheme.

Background

The proposed offline scheme uses two layers of chunks. Extending it to three or more layers would require compressing the second-layer emulator J while preserving sufficient structure. The authors state that when J is made subquadratic it is no longer faithful, and they do not know how to extract smaller emulators from it in subquadratic time.

References

In the second case, we do not know how to extract smaller emulators from $J$ in subquadratic time.

— Breaking Two Barriers for Dynamic Planar Reachability  (2610.13473 - Karczmarz et al., 9 Oct 2026) in Section 7, Conclusion, paragraph “Subpolynomial time”

In fact, it is unclear whether this problem can be solved even in polynomial time since we do not know how to reconstruct $G$ from~$H$ alone.

— Breaking Two Barriers for Dynamic Planar Reachability  (2610.13473 - Karczmarz et al., 9 Oct 2026) in Section 7, Conclusion, displayed Question