From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One 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 size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

/pdf/applications-of-random-sampling-in-computational-geometry-ii-3smdvp6m82.pdf

Applications of random sampling in computational geometry, II

An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ 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 $a \cdots b \cdots a \cdots b \cdots$ 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 wide variety of geometric problems can be formulated in terms of lower envelopes. A close to 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. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.

http://www.math.tau.ac.il/~michas/dssurvey.pdf

Davenport-Schinzel sequences and their geometric applications

Results of neighborhood graphs are surveyed. Properties, bounds on the size, algorithms, and variants of the neighborhood graphs are discussed. Numerous applications including computational morphology, spatial analysis, pattern classification, and databases for computer vision are described. >

Relative neighborhood graphs and their relatives

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.

/pdf/almost-optimal-set-covers-in-finite-vc-dimension-509wzhm9kv.pdf

Almost optimal set covers in finite VC-dimension

The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles. They also extend the analysis to three dimensions, where they concentrate on the case of spheres, which is relevant for the three-dimensional unit-distance problem. They obtain upper bounds for certain quantities. The authors believe that the techniques they use are of independent interest. >

Combinatorial Complexity Bounds for Arrangements of Curves and Surfaces

This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is $O(n\alpha (n)\log n)$. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.

Computing a face in an arrangement of line segments

LetS be a set ofn points in ?d. A setW is aweak ?-net for (convex ranges of)S if, for anyT⊆S containing ?n points, the convex hull ofT intersectsW. We show the existence of weak ?-nets of size % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaem4ta8KaeiikaGIaeiikaGIaeGymaeJaei4la8Ia% eqyTdu2aaWbaaSqabeaacqWGKbazaaGccqGGPaqkcyGGSbaBcqGGVb% WBcqGGNbWzdaahaaWcbeqaaiabek7aInaaBaaameaacqWGKbazaeqa% aaaakiabcIcaOiabigdaXiabc+caViabew7aLjabcMcaPiabcMcaPa% aa!50FE! $$O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))$$ , whereβ2=0,β3=1, andβd?0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/?) log1.6(1/?)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/?), which improves a previous bound of Capoyleas.

https://link.springer.com/content/pdf/10.1007%2FBF02574025.pdf

Improved bounds on weak ź-nets for convex sets

LetS be a set ofn points in ℝ d . A setW is aweak e-net for (convex ranges of)S if, for anyT⊆S containing en points, the convex hull ofT intersectsW. We show the existence of weak e-nets of size$O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))$, whereβ2=0,β3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/e) log1.6(1/e)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/e), which improves a previous bound of Capoyleas.

/pdf/improved-bounds-on-weak-e-nets-for-convex-sets-1ejeetc3gu.pdf

Leonidas J. Guibas

Papers

Combinatorial Complexity Bounds for Arrangements of Curves and Surfaces

Computing a face in an arrangement of line segments

Improved bounds on weak ź-nets for convex sets

Improved bounds on weak e-nets for convex sets

Topological Sweeping in Three Dimensions