Computing the Delaunay triangulation of n points is well known to have an Ω(n log n) lower bound. Researchers have attempted to break that bound in special cases where additional information is known. The Delaunay triangulation of the vertices of a convex polygon is such

Color red and blue the n vertices of a convex polytope P in R³. Can we compute the convex hull of each color class in o(n log n)? What if we have χ> 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.

Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the high-fidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Many interesting physical problems involve nonlinear and anisotropic behavior, and the PDEs modeling them exhibit discontinuities in their solutions. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary sim-plicial space domain. Our algorithm is the first spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena in anisotropic media using novel discontinuous Galerkin finite element methods for implicit solutions di-rectly in spacetime. Given a triangulated d-dimensional Euclidean space domain M (a simplicial complex) and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the (d + 1)-dimensional spacetime domain M × [0,∞). Our algorithm uses a near-optimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix M × [0,T] of spacetime. Unlike Delaunay meshes, the facets of our mesh satisfy gradient constraints that allow interleaving the construction of the mesh by adding new space-

From the early days of computational geometry, practitioners have looked for faster ways to triangulate a simple polygon. Several near-linear time algorithms have been devised and implemented. However, the true linear time algorithms of Chazelle and Amato et al. are considered impractical for actual use despite their faster asymptotic running time. In this paper, I examine the latter of these in detail and attempt to implement it. 1

The worst-case model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a three-dimensional convex polytope and a two-coloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offline-problem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the self-improving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a bounded-degree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.