Algorithms for distance problems in planar complexes of global nonpositive curvature

TL;DR

This paper introduces efficient algorithms for solving distance and convex hull problems in CAT(0) planar complexes, enabling fast shortest path queries and convex hull computations in such nonpositively curved spaces.

Contribution

It presents the first algorithms for single-point distance queries and convex hulls specifically in CAT(0) planar complexes, with proven efficiency.

Findings

Shortest path queries answered in linear time after O(n^2 log n) preprocessing

Convex hull computed in O(n^2 log n + nk log k) time

Data structures of size O(n^2) support these operations efficiently

Abstract

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex K with n vertices, one can construct in O(n^2 log n) time a data structure D of size O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n^2 log n + nk log k) time, using a data structure of size O(n^2 + k).