Algorithms for distance problems in planar complexes of global nonpositive curvature

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)

Geodesics in CAT(0) Cubical Complexes

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.Comment: 27 pages, 7 figure

Shortest path problem in rectangular complexes of global nonpositive curvature

International audienceCAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path gamma(x y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear polygons (CAT(0) rectangular complexes in which the links of all vertices are bipartite graphs) and squaregraphs (CAT(0) rectangular complexes arising from plane quadrangulations in which all inner vertices have degrees >= 4). Namely, we show that for a CAT(0) rectangular complex kappa with n vertices, one can construct a data structure D, of size O(n(2)) so that, given any two points x, y is an element of kappa, the shortest path gamma(x, y) between x and y can be computed in O (d(p, q)) time, where p and q are vertices of two faces of kappa containing the points x and y, respectively, such that gamma(x, gamma) subset of kappa(I(p, q)) and d(p, q) is the distance between p and q in the underlying graph of kappa. If kappa is a ramified rectilinear polygon, then one can construct a data structure D of optimal size O(n) and answer two-point shortest path queries in O(d(p, q)log Delta) time, where Delta is the maximal degree of a vertex of G(kappa). Finally, if kappa is a squaregraph, then one can construct a data structure D, of size O(n logn) and answer two-point shortest path queries in O(d(p, q)) time

The Markov-Dubins Problem with Free Terminal Direction in a Nonpositively Curved Cube Complex

State complexes are nonpositively curved cube complexes that model the state spaces of reconfigurable systems. The problem of determining a strategy for reconfiguring the system from a given initial state to a given goal state is equivalent to that of finding a path between two points in the state complex. The additional requirement that allowable paths must have a prescribed initial direction and minimal turning radius determines a Markov-Dubins problem with free terminal direction (MDPFTD). Given a nonpositively curved, locally finite cube complex X, we consider the set of unit-speed paths which satisfy a certain smoothness condition in addition to the boundary conditions and curvature constraint that define a MDPFTD. We show that this set either contains a path of minimal length, or is empty. We then focus on the case that X is a surface with a nonpositively curved cubical structure. We show that any solution to a MDPFTD in X must consist of finitely many geodesic segments and arcs of constant curvature, and we give an algorithm for determining those solutions to the MDPFTD in X which are CL paths, that is, made up of an arc of constant curvature followed by a geodesic segment. Finally, under the assumption that the 1-skeleton of X is d-regular, we give sufficient conditions for a topological ray in X of constant curvature to be a rose curve or a proper ray