A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes

TL;DR

This paper introduces the first polynomial time algorithm for computing geodesics in CAT(0) cubical complexes, utilizing an iterative approach based on a previous algorithm, applicable to general CAT(0) spaces.

Contribution

It provides a novel polynomial time algorithm for geodesic computation in CAT(0) cubical complexes, extending applicability to general CAT(0) spaces.

Findings

Algorithm successfully computes geodesics in polynomial time.

Applicable to any CAT(0) space with computable geodesics between close points.

Uses iterative breakpoint updates based on Miller, Owen, and Provan's method.

Abstract

This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Miller, Owen and Provan's algorithm (2015) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.