A generalization of the convex hull of a finite set of points in the plane leads to a family of straight-line graphs, "alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets.Expand

A randomized strategy for maintaining balance in dynamically changing search trees that has optimalexpected behavior, and generalizes naturally to weighted trees, where the expected time bounds for accesses and updates again match the worst-case time bounds of the best deterministic methods.Expand

An optimal algorithm is presented for constructing an arrangement of hyperplanes in arbitrary dimensions and is shown to improve known worst-case time complexities for five problems: computing all order-k Voronoi diagrams, computing the λ-matrix, estimating halfspace queries, degeneracy testing, and finding the minimum volume simplex determined by a set of points.Expand

We present an algorithm, APD, that solves the distance version of the all-pairs-shortest-path problem for undirected, unweighted n-vertex graphs in time O(M(n) log n), where M(n) denotes the time… Expand

We present a new planar convex hull algorithm with worst case time complexity $O(n \log H)$ where $n$ is the size of the input set and $H$ is the size of the output set, i.e. the number of vertices… Expand

It turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes, and this fact can be used to obtain new Vor onoi diagram algorithms.Expand

It is shown that disks and pseudo-disks in the plane as well as halfspaces in R<supscrpt>, which is best possible up to a multiplicative constant, allow-nets of size only &Ogr;(1/1/<italic>ε</italic)), which isbest possibleup to amultiplicative constant.Expand

Two randomized algorithms solve linear programs involvingm constraints ind variables in expected timeO(m) and constructs convex hulls ofn points in ℝd,d>3, in expectedTimeO(n[d/2]).Expand

The main tool in this new approach is the notion of a straight line shelling of a polytope in convex hull problems, which has best case time complexity O(m 2 -tFlogm), which is an improvement over the best previously achieved bounds for a large range of values of F.Expand