Linear or near-linear construction of the topological lunar EMST

Develop an algorithm that constructs the topological lunar Euclidean minimum spanning tree for n points with s+1 colors in O(n) time or, alternatively, O(n log n) time, without explicitly constructing the overlay of the s+1 Voronoi tessellations, whose size can be quadratic in n.

Background

The paper defines three versions of the lunar Euclidean minimum spanning tree: combinatorial, geometric, and topological. The topological version is obtained by canceling non-critical arcs and has at most O(n) vertices for n points and s+1 colors, as established by Lemma 2.?? (the paper's Lemma~\ref{lem:counting_critical_cells}).

The overlay of the s+1 Voronoi tessellations provides the basis for constructing the geometric lunar EMST but may have quadratic size. The unresolved algorithmic question is therefore whether the smaller topological tree can be constructed directly in linear or near-linear time while avoiding this potentially quadratic intermediate structure.

References

There are three versions of the lunar EMST, and the topological version for $n$ points and $s+1$ colors has at most $O(n)$ vertices; see Lemma~\ref{lem:counting_critical_cells}. Is there an algorithm that constructs this tree in time $O(n)$ or perhaps $O(n \log n)$? Such an algorithm would have to avoid the explicit construction of the overlay of the $s+1$ Voronoi tessellations, whose size can be quadratic in $n$.

— Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs  (2608.27118 - Draganov et al., 27 Aug 2026) in Section 5, Discussion

While $c_s$ is known to exist, we do not know its precise value, not even for $c_0$, which is the constant for the standard EMST. Lower and upper bounds as well as estimates from computational experiments for $c_1$ are given in ; see also Figure~\ref{fig:two-colors}. Extend these bounds and estimates to three or more colors. Falling short of pinning down the constants exactly, can we prove relations between them? For example, is it true that $c_s > c_{s+1}$ for all $s \geq 0$?

— Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs  (2608.27118 - Draganov et al., 27 Aug 2026) in Section 5, Discussion