Generalization to Higher-Dimensional Euclidean Spaces

Determine whether the techniques developed for minimum dilation trees in the Euclidean plane can be generalized to higher-dimensional Euclidean spaces, including whether an appropriate higher-dimensional analogue of the $r$-tree dissection can be established beyond the three-dimensional circle obstruction.

Background

The paper’s algorithm and structural analysis are developed specifically for point sets in the Euclidean plane. The conclusion asks whether the approach extends to higher dimensions and notes that even the basic rr-tree-dissection tool appears to require a stronger structural theorem than the available three-dimensional circle obstruction.

References

Thus, the open question raised by Eppstein of finding an approximation algorithm to compute a tree with minimum dilation remains open and consequently, this is open also for spanners of bounded tree-width.

— On (Directed) Width-Parameters of Geometric Spanners  (2609.22082 - Buchin et al., 18 Sep 2026) in Section Conclusion and Open Questions

A natural open question that arises from our results is: Can these techniques be generalized to higher dimensions? Even the $r$-tree-dissection, our most basic algorithmic tool, seems to require a stronger structural result than a three-dimensional version of the circle obstruction.

— A Sublinear Approximation Algorithm for Minimum Dilation Trees in the Plane  (2609.08990 - Berg et al., 8 Sep 2026) in Section 7, Section “Conclusion”