Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs

δ-Hyperbolic metric spaces have been defined by M. Gromov via a simple 4-point condition: for any four points <i>u,v,w,x</i>, the two larger of the sums <i>d</i>(<i>u,v</i>)+<i>d</i>(<i>w,x</i>), <i>d</i>(<i>u,w</i>)+<i>d</i>(<i>v,x</i>), <i>d</i>(<i>u,x</i>)+<i>d</i>(<i>v,w</i>) differ by at most 2δ. Given a finite set <i>S</i> of points of a δ-hyperbolic space, we present simple and fast methods for approximating the diameter of <i>S</i> with an additive error 2δ and computing an approximate radius and center of a smallest enclosing ball for <i>S</i> 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 <i>G</i>=(<i>V,E</i>) with uniformly bounded degrees of vertices, the exact center of <i>S</i> can be computed in linear time <i>O</i>(|E|). We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs <i>G</i> on <i>n</i> vertices with an additive error <i>O</i>(δlog₂ <i>n</i>). This construction has an additive error comparable with that given by Gromov for <i>n</i>-point δ-hyperbolic spaces, but can be implemented in <i>O</i>(|E|) time (instead of <i>O</i>(<i>n</i>²)). Finally, we establish that several geometrical classes of graphs have bounded hyperbolicity.