Reusability of ICALP 2025 Voronoi data structures across recursive levels

Determine whether the data structure required by Boneh, Golan, Mozes, Prigan, and Weimann (ICALP 2025) for O(S)-time construction of additively weighted Voronoi diagrams in undirected planar graphs can be efficiently reused across recursive levels, so that recursive Voronoi-based distance oracles avoid incurring O(|H|) preprocessing time at each level.

Background

Recent work (ICALP 2025) achieved O(S)-time construction of additively weighted Voronoi diagrams in undirected planar graphs after an O(|H|) preprocessing step by employing a binary search over shortest-path separators. However, that approach relies on a data structure whose construction time is linear in the size of the region, which prevents combining it with recursive distance oracles that cannot afford linear preprocessing at each recursion level.

A natural idea is to reuse data structures built at one recursion level in subsequent levels to amortize or eliminate the per-level linear preprocessing. Whether this reuse strategy applies to the specific data structure required by the ICALP 2025 method remains unresolved.

References

A possible solution is to devise a way to efficiently reuse the data structures already computed for one level in subsequent recursive levels, but it is unknown whether this idea is applicable to the data structure required in.

— Distances in Planar Graphs are Almost for Free!  (2603.26313 - Mozes et al., 27 Mar 2026) in Introduction — Voronoi Diagrams subsection (discussion of ICALP 2025 approach)