Maximum spanning trees in normed planes

TL;DR

This work extends the Monma–Paterson–Suri–Yao approach to maximum spanning trees from the Euclidean plane to general normed planes $\mathbb{M}^2$, leveraging furthest Voronoi/abstract Voronoi structures to obtain an $O(n\log n)$-time construction of the maximum spanning tree under distinct distances. It analyzes the geometric structure via the furthest neighbour graph $\mathtt{FNG}(S)$, showing that MXST edges either belong to $\mathtt{FNG}(S)$ or join adjacent components in the cyclic order around $\partial\mathrm{conv}(S)$, and provides a concrete algorithm with time bounds dependent on the availability of furthest diagram subroutines. The paper also demonstrates applications to the 2-clustering problem (minimizing maximum diameter) with provable runtimes, and offers a perturbation method for strictly convex norms to ensure all distances are distinct, enabling direct application of the MXST algorithm. It also clarifies the limitations of such perturbations in non-strictly convex planes, highlighting the structural differences across normed geometries.

Abstract

Extending some properties from the Euclidean plane to any normed plane, we show the validity of the Monma-Paterson-Suri-Yao algorithm for finding the maximum-weighted spanning tree of a set of $n$ points, where the weight of an edge is the distance between the end points measured by the norm and there are not repeated distances. For strictly convex normed planes, we expose an strategy for moving slightly the points of the set in order to obtain distinct distances.

Figures (4)

• Figure 1: $\mathtt{MXST}(S)$ of a set $S$ of $n$ points in $\mathbb{E}^2$.
• Figure 2: $\overline{xy}\notin \mathtt{MXST}(S)$ if $x,y$ are not on $\partial\mathrm{conv}(S)$.
• Figure 3: The construction of $\mathtt{MXST}(S)$.
• Figure 4: $\|p_1-p_2\|=\|p_1-p_3\|=1$, $\|p'_1-p'_2\|=\|p'_1-p'_3\|=1.1$, with $p_i'=p_i+\epsilon_i$ ($i=1,2,3)$ and $\epsilon_1<\epsilon_2<\epsilon_3$.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4