Maximum spanning trees in normed planes

TL;DR

This paper extends properties of maximum spanning trees from Euclidean to general normed planes, validating an algorithm and proposing a method for strictly convex norms to ensure distinct distances.

Contribution

It generalizes the maximum spanning tree algorithm to all normed planes and introduces a point-moving strategy for strictly convex norms to avoid repeated distances.

Findings

The Monma-Paterson-Suri-Yao algorithm is valid in any normed plane.

A strategy is provided to ensure distinct distances in strictly convex normed planes.

The approach broadens the applicability of maximum spanning tree algorithms beyond Euclidean geometry.

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.