Shortest Paths and Convex Hulls in 2D Complexes with Non-Positive Curvature

TL;DR

This paper develops algorithms for computing convex hulls and shortest path queries in 2D CAT(0) polyhedral complexes, spaces with unique shortest paths that are relevant in biology and robotics, extending Euclidean methods.

Contribution

It introduces a linear programming algorithm for convex hulls and unifies shortest path solutions in 2D CAT(0) complexes, addressing a gap in non-Euclidean geometry algorithms.

Findings

Linear programming computes convex hulls in 2D CAT(0) complexes.

Shortest path maps enable efficient single-source queries.

Unified approach for 2-manifold and rectangular cases.

Abstract

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensional CAT(0) polyhedral complexes, and we unify efficient solutions for 2-manifold and rectangular cases.