In this paper, we present algorithms for computing approximate hulls and centerpoints for collections of matrices in positive definite space. There are many applications where the data under consideration, rather than being points in a Euclidean space, are positive definite (p.d.) matrices. These applications include diffusion tensor imaging in the brain, elasticity analysis in mechanical engineering, and the theory of kernel maps in machine learning. Our work centers around the notion of a horoball: the limit of a ball fixed at one point whose radius goes to infinity. Horoballs possess many (though not all) of the properties of halfspaces; in particular, they lack a strong separation theorem where two horoballs can completely partition the space. In spite of this, we show that we can compute an approximate "horoball hull" that strictly contains the actual convex hull. This approximate hull also preserves geodesic extents, which is a result of independent value: an immediate corollary is that we can approximately solve problems like the diameter and width in positive definite space. We also use horoballs to show existence of and compute approximate robust centerpoints in positive definite space, via the horoball-equivalent of the notion of depth.
For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space. Algorithms for constructing these triangulations will also be described. We initiate the study of approximate algorithms on negatively curved spaces. These spaces have recently become of interest in various domains of computer science including networking and vision. The classical example of such a space is the real-hyperbolic space H ^d for d 2, but our approach applies to a more general family of spaces characterized by Gromov's (combinatorial) hyperbolic condition. We give efficient algorithms and data structures for problems like approximate nearest-neighbor search and compact, low-stretch routing on subsets of negatively curved spaces of fixed dimension (including H ^d as a special case). In a different direction, we show that there is a PTAS for the Traveling Salesman Problem when the set of cities lie, for example, in H ^d. This generalizes Arora's results for R ^d. Most of our algorithms use the intrinsic distance geometry of the data set, and only need the existence of an embedding into some negatively curved space in order to function properly. In other words, our algorithms regard the interpoint distance function as a black box, and are independent of the representation of the input points. We provide efficient constant-factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces. δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points u,v,w,x, the two larger of the sums d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w) differ by at most 2δ. Given a finite set S of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of S with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for S with an additive error 3δ. These algorithms run in linear time for classical hyperbolic spaces and for δ-hyperbolic graphs and networks. Furthermore, we show that for δ-hyperbolic graphs G=(V,E) with uniformly bounded degrees of vertices, the exact center of S can be computed in linear time O(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δlog2 n). This construction has an additive error comparable with that given by Gromov for n-point δ-hyperbolic spaces, but can be implemented in O(|E|) time (instead of O(n2)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity. We consider the problem of covering and packing subsets ofΔ-hyperbolic metric spaces and graphs by balls.These spaces, defined via a combinatorial Gromov condition, haverecently become of interest in several domains of computer science.Specifically, given a subset Sof aΔ-hyperbolic graph Gand a positive numberR, let Δ(S,R) be theminimum number of balls of radius Rcovering S.It is known that computing Δ(S,R)or approximating this number within a constant factor is hard evenfor 2-hyperbolic graphs. In this paper, using a primal-dualapproach, we show how to construct in polynomial time a covering ofSwith at most Δ(S,R)balls of (slightly larger) radius R+ Δ.This result is established in the general framework ofΔ-hyperbolic geodesic metric spaces and is extendedto some other set families derived from balls. The coveringalgorithm is used to design better approximation algorithms for theaugmentation problem with diameter constraints and for thek-center problem in Δ-hyperbolicgraphs.