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.

We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...

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...

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 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

Let be a set of points in in general position, i.e., no points on a common-flat,. A-set of is a set of points in that can be separated from by a hyperplane. A-facet of is an oriented-simplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar point set and is even, a halving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number! "$ # of-sets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number of-sets we show that

INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of `physical world' application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in low-dimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we in

Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

Introduction Let S be a set of n points in general position in the plane, i.e. no three points are collinear. Four points in S may or may not form the vertices of a convex quadrilateral; if they do, we call this subset of 4 elements convex. We are interested in the number of convex 4-element subsets. This can of course be as large as , if S is in convex position, but what is its minimum? Another way of stating the problem is to find the rectilinear crossing number of the complete n-graph K n , i.e., to determine the minimum number of crossings in a drawing of K n in the plane with straight edges and the nodes in general position. We note here that the rectilinear crossing number of complete graphs also determines the rectilinear crossing number of random graphs (provided the probability for an edge to appear is at least ln n ), as was shown by Spencer and Toth [13]. It is easy to see that for n = 5 we get at least one convex 4-element subset, from which it follows by straigh

Abstract — We propose a low overhead scheme for detecting a network partition or cut in a sensor network. Consider a network S of n sensors, modeled as points in a two-dimensional plane. An ε-cut, for any 0 < ε < 1, is a linear separation of εn nodes in S from a distinguished node, the base station. We show that the base station can detect whenever an ε-cut occurs by monitoring the status of just O ( 1) nodes in the network. ε Our scheme is deterministic and it is free of false positives: no reported cut has size smaller than 1 εn. Besides this combinatorial result, we also 2 propose efficient algorithms for finding the O ( 1) nodes that should act ε as sentinels, and report on our simulation results, comparing the sentinel algorithm with two natural schemes based on sampling. I.