A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes

Koyo Hayashi

doi:10.1007/s00454-019-00154-2

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 Owen and Provan’s algorithm (IEEE/ACM Trans Comput Biol Bioinform 8(1):2–13, 2011) as a subroutine. Our algorithm is applicable to any path in any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.

Highlights

Computing a shortest path in a polyhedral domain in Euclidean space is a fundamental and important algorithmic problem, which is intensively studied in computational geometry [16]
There are some cases where one can obtain polynomial time complexity. It was shown by Sharir [24] that a shortest obstacle-avoiding path among k disjoint convex polyhedra having altogether n vertices, can be found in nO(k) time, which implies that this problem is polynomially solvable if k is a small constant
Owen and Sullivant [2] gave a combinatorial description of CAT(0) cubical complexes, employing a poset endowed with an additional relation, called a poset with inconsistent pairs (PIP). This can be viewed as a generalization of Birkhoff’s theorem that gives a compact representation of distributive lattices by posets. They showed that there is a bijection between CAT(0) cubical complexes and PIPs. (Through the abovementioned equivalence, this can be viewed as a rediscovery of the result of Barthélemy and Constantin [4], who found a bijection between PIPs and pointed median graphs.) This relationship enables us to express an input CAT(0) cubical complex as a PIP: For a poset with inconsistent pairs P, the corresponding CAT(0) cubical complex KP is realized as a subcomplex of the |P |-dimensional cube [0, 1]P in which the cells of KP are specified by structures of P

Summary

Introduction

Computing a shortest path in a polyhedral domain in Euclidean space is a fundamental and important algorithmic problem, which is intensively studied in computational geometry [16]. (Through the abovementioned equivalence, this can be viewed as a rediscovery of the result of Barthélemy and Constantin [4], who found a bijection between PIPs and pointed median graphs.) This relationship enables us to express an input CAT(0) cubical complex as a PIP: For a poset with inconsistent pairs P , the corresponding CAT(0) cubical complex KP is realized as a subcomplex of the |P |-dimensional cube [0, 1]P in which the cells of KP are specified by structures of P Adopting this embedding as an input, they gave the first algorithm to compute geodesics in an arbitrary CAT(0) cubical complex.

Algorithm

Analysis

Median graph

Main theorem