Rooted Uniform Monotone Minimum Spanning Trees

TL;DR

This paper develops algorithms for constructing rooted minimum spanning trees with monotonicity constraints, focusing on y-monotonicity and xy-monotonicity, with efficient computation when directions are known or need to be optimized.

Contribution

It introduces algorithms for computing rooted monotone minimum spanning trees efficiently, including methods to determine optimal directions for monotonicity.

Findings

Algorithms run in O(|P| log^2 |P|) and O(|P| log^3 |P|) time for known directions.

Optimal directions for monotonicity can be found in O(|P|^2 log |P|) time.

Deciding if all paths from root are monotone w.r.t. a direction is also addressed.

Abstract

We study the construction of the minimum cost spanning geometric graph of a given rooted point set $P$ where each point of $P$ is connected to the root by a path that satisfies a given property. We focus on two properties, namely the monotonicity w.r.t. a single direction ($y$-monotonicity) and the monotonicity w.r.t. a single pair of orthogonal directions ($xy$-monotonicity). We propose algorithms that compute the rooted $y$-monotone ($xy$-monotone) minimum spanning tree of $P$ in $O(|P|\log^2 |P|)$ (resp. $O(|P|\log^3 |P|)$) time when the direction (resp. pair of orthogonal directions) of monotonicity is given, and in $O(|P|^2\log|P|)$ time when the optimum direction (resp. pair of orthogonal directions) has to be determined. We also give simple algorithms which, given a rooted connected geometric graph, decide if the root is connected to every other vertex by paths that are all monotone w.r.t. the same direction (pair of orthogonal directions).