An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

... In this paper we present a general framework for addressing such problems and the tools for designing and analyzing relevant algorithms, which we call kinetic data structures. We discuss kinetic data structures for a variety of fundamental geometric problems, such as the maintenance of convex hulls, Voronoi and Delaunay diagrams, closest pairs, and intersection and visibility problems. We also briefly address the issues that arise in implementing such structures robustly and efficiently. The resulting techniques satisfy three desirable properties: (1) they exploit the continuity of the motion of the objects to gain efficiency, (2) the number of events processed by the algorithms is close to the minimum necessary in the worst case, and (3) any object may change its `flight plan' at any moment with a low cost update to the simulation data structures. For computer applications dealing with motion in the physical world, kinetic data structures lead to simulation performance unattainable by other means. In addition, they raise fundamentally new combinatorial and algorithmic questions whose study may prove fruitful for other disciplines as well.

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...

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R^d, the maximum complexity of its Voronoi diagram under the L1 metric, and also under a simplicial distance function, are both shown to be \Theta(n dd=2e ). The upper bound for the case of the L1 metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R^d. This complexity is \Theta(n dd=2e ), for d 1, and it improves to \Theta(n bd=2c ), for d 2, if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R³ is shown to be \Theta(n 2 ). We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in d...

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the k-level of the arrangement A(F) is O(k 3+" /(n=k)), for any " ? 0, where /(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F . This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [51], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static...

We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteed-quality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionality---most notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20 # , as measured from the skewed perspective of any point in the triangle.

. We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3-space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k 3 + kn log k log n) expected time. Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539793250755 1. Combinatorial bounds. Let P = {P 1 , . . . , P k } be a family of k convex polyhedra in 3-space, let n i be the number of faces of P i , and let n = # k i=1 n i . Put U = # P. By the combinatorial complexity of a polyhedral set we mean the total number of its vertices, edges, and faces. Our main result is the followin...

We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of n low-degree algebraic surface patches in 3-space. We show that this complexity is O(n