Geodesics in CAT(0) Cubical Complexes

TL;DR

This paper presents an algorithm for computing geodesics in CAT(0) cubical complexes, leveraging a correspondence with posets to facilitate explicit realizations and embeddings.

Contribution

It introduces a novel algorithm for geodesic computation in CAT(0) cubical complexes based on a new correspondence with posets with inconsistent pairs.

Findings

Algorithm efficiently computes geodesics in CAT(0) cubical complexes.

Provides explicit realizations of complexes as state complexes.

Enables embedding of intervals into integer lattice cubings.

Abstract

We describe an algorithm to compute the geodesics in an arbitrary CAT(0) cubical complex. A key tool is a correspondence between cubical complexes of global non-positive curvature and posets with inconsistent pairs. This correspondence also gives an explicit realization of such a complex as the state complex of a reconfigurable system, and a way to embed any interval in the integer lattice cubing of its dimension.