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 describe the following data structures. For halfspace range reporting, in 3-space using expected preprocessing time O(n log n), worst-case storage O(n log log n) and worst-case reporting time O(log n + k), where n is the number of data points and k the number of points reported; in d-space, with d even, using worst-case preprocessing time O(n log n), storage O(n) and reporting time O(n 1 1=bd=2c log c n + k), where c is a constant. For ray shooting in a convex polytope in d-space determined by n facets, using deterministic preprocessing time O((n= log n) bd=2c log c n) and storage O((n= log n) bd=2c 2 c log n ) and with query time O(log n). For ray shooting in arbitrary direction among n hyperplanes using preprocessing O(n d = log bd=2c n) and query time O(log n). We also describe a randomized algorithm for constructing the k-level of n planes in 3-space. In the case of planes dual to points in convex position, in which the size of the k-level is O(nk), the a...

Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space, deciding whether it admits a crossing-free projection can be done in O(n 2 log n+k) time and O(n 2 +k) space, where k is the number of edge intersections of forbidden quadrilaterals (i.e. set of directions that admits a crossing) and varies from zero to O(n 4 ). This implies for example that given a simple polygon in 3-space we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, a minimum-crossing projection can be found in O(n 4 ) time and space. We show that an object always admits a regular projection (of interest to k...

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...

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

We consider the problem of computing the lower envelope (the minimum) of n constant degree algebraic functions of one variable. The lower envelope has size O(nfi(n)) where fi(n) is a nearly constant function, and it can easily be computed in time O(nfi(n) log n) by a simple deterministic divide-and-conquer algorithm [45]. We give an alternative simple (module a derandomization black box) approach using divide-and-conquer based on cuttings that results in a deterministic sequential algorithm that runs in the same time bound. This algorithm uses derandomization tools by now standard. This approach however allows us to obtain the following results: ffl A deterministic sequential algorithm that is output sensitive and runs in time O(n log f) if f n ffl , or O(nfi(f) log f) = O(nfi(n) log n) otherwise, where f is the size of the output; ffl a randomized parallel EREW algorithm that runs in time O(log n) and uses nearly optimal work O(nfi 2 (n) log n) with n-polynomial probability...

We describe two deterministic algorithms for constructing the arrangement determined by a set of (algebraic) curve segments in the plane. They both use a divide-and-conquer approach based on derandomized geometric sampling and achieve the optimal running time O(n log n + k), where n is the number of segments and k is the number of intersections. The rst algorithm, a simpli ed version of one presented in [1], generates a structure of size O(n log log n + k) and its parallel implementation runs in time O(log 2 n). The second algorithm is better in that the decomposition of the arrangement constructed has optimal size O(n + k) and it has a parallel implementation in the EREW PRAM model that runs in time O(log 3=2 n). The improvements in the second algorithm are achieved by means of an approach

We give fast and efficient methods for constructing... time using linear work on an EREW PRAM.

We show that several discrepancy-like problems can be solved in NC 2 nearly achieving the corresponding sequential bounds. For example, given a set system (X; S), where X is a ground set and S ` 2 X , a set R ` X can be computed in NC 2 so that, for each S 2 S, the discrepancy jjR " Sj \Gamma jR " Sjj is O( p jSj log jSj). Previous NC algorithms could only achieve O( p jSj 1+ffl log jSj), while ours matches the probabilistic bound achieved sequentially by the method of conditional probabilities within a multiplicative factor 1 + o(1). Other problems whose NC solution we improve are lattice approximation, ffl-approximations of range spaces of bounded VC-exponent, sampling in geometric configuration spaces, and approximation of integer linear programs. 1 Introduction Problem and previous work. Discrepancy is an important concept in combinatorics, see e.g. [1, 5], and theoretical computer science, see e.g. [27, 23, 9]. It attempts to capture the idea of a good sample from ...

We investigate the I/O-complexity of computing the trapezoidal decomposition defined by a set of N line segments in the plane. We present a randomized algorithm which solves optimally this problem requiring O( N B log M=B N B + K B ) expected I/O operations, where K is the number of pairwise intersections, M is the size of available internal memory and B is the size of the block transfer. The proposed algorithm requires an optimal expected number of internal operations. As a by-product, the algorithm also solves the segment intersections problem requiring the same number of I/Os and internal operations.