The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

We present an efficient O(n + 1/ε^4.5)-time algorithm for computing a (1 + 1/ε)-approximation of the minimum-volume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online [Har00].

We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under translation, computing rectilinear p-centers and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm rand-min 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a well-known fact [27, 44], the expected number of times that step 5 is execut...

Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2-d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3-d runs in near O(n + k ) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the ( k)-level, previously used in proving combinatorial k-level bounds.

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangent-finding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...

We derive two results from Clarkson's randomized algorithm for linear programming in a fixed dimension d. The first is a simple general method that reduces the problem of answering linear programming queries to the problem of answering halfspace range queries. For example, this yields a randomized data structure with O(n) space and O(n 1\Gamma1=bd=2c 2 O(log n) ) query time for linear programming on n halfspaces (d ? 3). The second result is a simpler proof of the following: a sequence of q linear programming queries on n halfspaces can be answered in O(n log q) time, if q n ff d for a certain constant ff d ? 0. Unlike previous methods, our algorithms do not require parametric searching. 1 Introduction One of the major discoveries in computational geometry is that fixed-dimensional linear programming can be solved in linear time [Meg84]. It was observed that the introduction of randomization leads to considerably simpler solutions [Sei91, Cla95]. The goal of this paper is...

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the order-k Voronoi diagram in O(n log n + nk log k) expected time.

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean d-dimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...

We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.