Match the quadratic preprocessing bound for hyperbolic shortest paths

Determine whether the O(g^2) preprocessing time of the Chen–Han shortest-path query structure can be matched for shortest-path computation on a smooth hyperbolic surface represented by a Dirichlet domain.

Background

The paper adapts continuous-Dijkstra and windowing techniques originally developed for shortest paths on polyhedral surfaces. Chen and Han’s method achieves O(n2) preprocessing by exploiting planar unfolding and curvature concentrated at vertices. Those structural properties do not hold on a smooth hyperbolic surface, where curvature is distributed uniformly and the surface cannot be treated through the same unfolding framework.

The authors obtain an O(g2) algorithm for computing a single distance on a genus-g hyperbolic surface, but explicitly leave unresolved whether the corresponding quadratic preprocessing bound can also be achieved in their setting for shortest-path queries.

References

Whether their $O(n2)$ preprocessing time can be matched in our setting is an open problem.

— The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems  (2609.09549 - Despre et al., 9 Sep 2026) in Section 2, subsection “The discrete geodesic problem”