Algorithms for Distance Problems in Planar Complexes of Global Nonpositive Curvature

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 ∈ K, the shortest path γ(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).

Shortest paths and convex hulls in 2D complexes with non-positive curvature

“…We first show that the shortest path map may have exponential size for a general 2D CAT(0) complex. This contrasts with the fact that the shortest path map has size O(n 2 ) in the two special cases where the single-source shortest path problem is known to be efficiently solvable: when the complex is a topological 2-manifold with boundary, which we will call a 2-manifold for short [41]; and when the complex is rectangular [19].…”

“…If the complex is a 2-manifold (i.e., every edge is in at most two faces) then no bifurcations can occur, so each boundary tree consists of only one branch, which implies that the size of the shortest path map is O(n 2 ). This was proved by Maftuleac [41] (where 2-manifold complexes are called "planar"), but we include a proof because we wish to observe a generalization. Lemma 10 ( [41]).…”

“…For 2D CAT(0) complexes that are 2-manifolds, the algorithm of Chen and Han [16] applies (in fact, they do not need that CAT(0) property), and solves the single-source shortest path problem with preprocessing time O(n 2 ), space O(n) and query time (to produce the path) proportional to the size of the output path. Maftuleac [41] explicitly discussed this as a problem in a CAT(0) space and gave a Dijkstra-like algorithm with the same space and query time, but preprocessing time of O(n 2 log n). Chepoi and Maftuleac [19] used different methods to give a polynomial time algorithm for all-pair shortest path queries in any 2D CAT(0) rectangular complex, with preprocessing time O(n 2 ), space O(n 2 ), and query time proportional to the size of the output path.…”

“…However, in a polyhedral surface with unique geodesics the convex hull is well defined, and Maftuleac [41] gave an algorithm to compute the convex hull of a set of points in O(n 2 log n) time, where n is the number of vertices in the complex plus the number of points in the set.…”