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.

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 consider the problem of bounding the complexity of the k-th level in an arrangement of n curves or surfaces, a problem dual to, and extending, the well-known k-set problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n p k + 1) for the complexity of the k-th level in an arrangement of n lines. (b) We derive an improved version of Lov'asz Lemma in any dimension, and use it to prove a new bound, O(n 2

An extremal problem considering sequences related to Davenport-Schinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x i-times).

We investigate 3-d visibility problems in which the viewing position moves along a straight flightpath. Specifically we focus on two problems: determining the points along the flightpath at which the topology of the viewed scene changes, and answering ray-shooting queries for rays with origin on the flightpath. Three progressively more specialized problems are considered: general scenes, terrains, and terrains with vertical flightpaths. 1.

. We present a simple but powerful new probabilistic technique for analyzing the combinatorial complexity of various substructures in arrangements of piecewise-linear surfaces in higher dimensions. We apply the technique (a) to derive new and simpler proofs of the known bounds on the complexity of the lower envelope, of a single cell, or of a zone in arrangements of simplices in higher dimensions, and (b) to obtain improved bounds on the complexity of the vertical decomposition of a single cell in an arrangement of triangles in 3-space, and of several other substructures in such an arrangement (the entire arrangement, all nonconvex cells, and any collection of cells). The latter results also lead to improved algorithms for computing substructures in arrangements of triangles and for translational motion planning in three dimensions. 1. Introduction The study of arrangements of curves or surfaces is an important area of research in computational and combinatorial geometry, because many...

. Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 0-1 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function. 1. Introduction A configuration, C = (c ij ) (1 i u; 1 j v), is a partial matrix with 1's and blanks at the entries. All the matrices we are going to work with will be 0 \Gamma 1 matrices. We say that a matrix M = (m ij ) does have the configuration C if one can find u rows i 1 ; i 2 ; . . . ; i u ; i 1 ! \Delta \Delta \Delta ! i u and v columns j 1 ; j 2 ; . . . ; j v ; j 1 ! \Delta \Delta \Delta ! j v in M such that the corresponding submatrix contains C, i.e. m i ff ;j fi = 1 whenever c ff;fi = 1. Let f(n; m;C) denote the maximum number of 1's in an n \Theta m matrix M not containing C. In the case of n = m we writ...

The combinatorial complexity of the Voronoi diagram of n lines in three dimensions under a convex distance function induced by a polytope with a constant number of edges is shown to be O(n 2 ff(n) log n), where ff is a slowly growing inverse of the Ackermann function. There are arrangements of n lines where this complexity can be as large as \Omega\Gamma n 2 ff(n)). 1 Introduction Statement of result. Let P be a closed convex polytope in 3-space which contains the origin. Given any point w 2 IR 3 , its distance from a line (or any other object) `, as induced by P , is d P (w; `) = inf ft 0 : (w + tP ) " ` 6= ;g ; Work by Paul Chew and Klara Kedem has been supported by AFOSR Grant AFOSR-91-0328 and by the U.S.--Israeli Binational Science Foundation. Work by Paul Chew has also been supported by ONR Grant N00014-89-J-1946 and ARPA under ONR contract N00014-88-K-0591. Work by Micha Sharir and Emo Welzl has been supported by the G.I.F. --- the German Israeli Foundation for Sc...

We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: 1. The maximum complexity of the entire common exterior is \Theta(nff(k) + k 2 ). 1 2. The maximum complexity of a single cell of the common exterior is \Theta(nff(k)). 3. The complexity of m distinct cells in the common exterior is O(m 2=3 k 2=3 log 1=3 ( k 2 m )+ n log k) and can be \Omega\Gamma m 2=3 k 2=3 + nff(k)) in the worst case. 1 Introduction In this paper we establish several combinatorial bounds on the complexity of the common exterior (namely, the complement of the union) of a collection of k convex polygons in the plane, with a total of n edges. The arrangement of such a collection of polygons can be viewed as a special case of an arrangement of n segments, but we prefer to regard it as a generalization of an arrangement of k...

The problem of bounding the combinatorial complexity of a single connected component (a single cell) of the complement of a set of n geometric objects in R k of constant description complexity is an important problem in computational geometry which has attracted much attention over the past decade. It has been conjectured that the combinatorial complexity of a single cell is bounded by a function much closer to O(n k 1 ) rather than O(n k ) which is the bound for the combinatorial complexity of the whole arrangement. Till now, this was known to be true only for k 3 and only for some special cases in higher dimensions. A classic result in real algebraic geometry due to Oleinik-Petrovsky, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. However, till now no better bounds were known if we restricted attention to a single connected component of a basic semi-algebraic set. In this paper, we show how these two problems...