We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d. A point c ∈ IR d is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d + 1)-center point; our algorithm finds an Ω(1/d 2)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log 2 d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ɛ-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly. 1

We give fast randomized and deterministic parallel meth-ods for constructing convex hulls in IR d, for any fixed d. Our methods are for the weakest shared-memory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In partic-ular, we show that the convex hull of n points in IRd canbe constructed in O(log n) time using O(n log n + nbd=2c)work, with high probability. We also show that it can be constructed deterministically in O(log2 n) time using O(n log n) work for d = 3 and in O(log n) time using O(nbd=2c logc(dd=2e\Gamma bd=2c) n) work, for d * 4, where c? 0is a constant, which is optimal for even d * 4. We also showhow to make our 3-dimensional methods output-sensitive with only a small increase in running time.These methods can be applied to other problems as well. A variation of the convex hull algorithm for even dimen-sions deterministically constructs a (1=r)-cutting of n hy-perplanes in IR d in O(log n) time using optimal O(nrd\Gamma 1) work; when r = n, we obtain their arrangement and a pointlocation data structure for it. With appropriate modifications, our deterministic 3-dimensional convex hull algorithmcan be used to compute, in the same resource bounds, the intersection of n balls of equal radius in R³. This leads to asequential algorithm for computing the diameter of a point set in IR3 with running time O(n log³ n), which is arguablysimpler than an algorithm with the same running time by Brönnimann et al.

We define two results on approximate shortest path maps in IR 3 . (i) Given a polyhedral surface or a convex polytope P with n edges in IR 3 , a source point s on P , and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision of P of size O((n=") log(1=")) which can be used to answer efficiently approximate shortest path queries. Namely, given any point t on P , one can compute, in O(log (n=")) time, a distance \Delta P;s (t), such that dP;s (t) \Delta P;s (t) (1 + ")d P;s (t), where dP;s (t) is the length of a shortest path between s and t on P . The map can be computed in O(n 2 log n + (n=") log (1=") log (n=")) time, for the case of a polyhedral surface, and in O((n=" 3 ) log(1=") + (n=" 1:5 ) log (1=") log n) time if P is a convex polytope. (ii) Given a set of polyhedral obstacles O with a total of n edges in IR 3 , a source point s in IR 3 n int S O2O O, and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision o...

We present memory-efficient deterministic algorithms for constructing #-nets and #-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.

Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in three-dimensional space. Our main results include: 1. A tight �(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of �(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for anyε>0, that builds a structure supporting O(log n)-time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the “towering property ” in O(n4/3+ε) time, for any ε>0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and ε-nets for various geometric range spaces.

We prove a general lemma on the existence of (1/r)-cuttings of geometric objects in I = that stisfy certain properties. We use this lemma to construct (1/r)-cuttings of (azymptotically) optimal size for arrangements of line segments in the plane and arrangements of triangles in 3-space; for line segments in the plane we obtain a cutting of size O(r + Ar2/n2), and for triangles in 3-space our cutting haz size O(r2(r) + Ara/nZ). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3-spce.

Using a divide, prune, and conquer approach based on geometric partitioning, we obtain: (1) An output sensitive algorithm for computing a weighted Voronoi diagram in R 4 (the projection of certain polyhedra in R 5) that runs in time O((n+f) log 3 f) where n is the number of sites and f is the number of output cells; and (2) a deterministic parallel algorithm in the EREW PRAM model for computing an algebraic planar Voronoi diagram (in which bisectors between sites are simple curves consisting of a constant number of algebraic pieces of constant degree) that runs in time O(log 2 n) using optimal O(n log n) work. The first result implies an algorithm for the problems of computing the convex hull of a point set and the intersection of a set of halfspaces in R 5, and computing the Euclidean Voronoi diagram in R 4. The second result implies both sequential and parallel work-optimal deterministic algorithms for a number of Voronoi diagram problems (including line segments in the plane), and other non-Voronoi diagram problems that can fit in the framework (including the intersection of equal radius balls in R 3 and some lower envelope problems in R 3). 1

Consider a point set D with a measure functionµ: D→R. Let A be the set of subsets of D induced by containment in a shape from some geometric family (e.g. axis-parallel rectangles, half planes, balls, k-oriented polygons). We say a range space (D, A) has anε-approximation P if

We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an approximate center point of a set S ∈ R d with running time sub-exponential in d. The algorithm is a derandomization of the Iterated-Radon algorithm of Clarkson et al and is guaranteed to terminate with an O(1/d 2)-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-Completenes of testing center points in general. We also explore the use of higher order Tverberg partitions to improve the runtime of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the O(1/d2) center of the Iterated-Radon algorithm to O(1/d r r−1) for a cost of O((rd) d) in time for any integer r. 1

Discrepancy theory investigates how uniform nonrandom structures can be. For example, given n points in the plane, how should we color them red and blue so as to minimize the difference between the number of red points and the number of blue ones within any disk? Or, how should we place n points in the unit square