Combinatorial Problems in Computational Geometry

In this thesis we study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions. Some of these problems have algorithmic applications, while others provide combinatorial bounds for various structures in such arrangements. The thesis involves two main themes: (i) Counting Crossing Configurations in Geometric Settings and its Applications: Suppose we “draw” a simple undirected graph G = (V, E) in the plane using points to represent vertices, and Jordan arcs connecting them to represent edges. Assume that G has n vertices and m edges and that m ≥ 4n. Then, using a planarity argument, there must exist two crossing arcs in this drawing. This fact can be exploited to show that the number of such crossings is Ω(m/n), no matter how the graph is drawn. The proof of this “Crossing Lemma” is due to Leighton [Lei83] and to Ajtai et al. [ACNS82]. A probabilistic proof of this fact was entitled “A proof from the book” [AZ98]. Adapting and extending the proof technique of the Crossing Lemma, we provide improved asymptotic bounds on well-studied geometric combinatorial problems, such as the “k-set” problem (Chapter 4), the complexity of polytopes spanned by sets of points in the plane and in space (Chapter 3), etc. In Chapter 2 we provide some sharp asymptotic Ramsey type theorems for intersection patterns of “nice” objects that are spanned by finite point sets: For example, we prove that for any dimension d, there exists a constant c = c(d) such that for any set P of n points in IR and any set S of m > cn distinct balls, each bounded by a sphere passing through a distinct pair of points of P , there exists a subset S ′ of S of size at least Ω(m/n) with nonempty intersection. This is asymptotically tight and improves the previously best known bound (see [CEG94]). We extend this result to other families of objects, including pseudo-disks in the plane and axis-parallel boxes in any dimension. The proofs rely on the same probabilistic proof technique of the Crossing Lemma, and can be regarded as extensions of that lemma. The results of this chapter are joint work with Micha Sharir and appear in [SS03b]. In Chapter 3 we prove that the maximum total complexity of k non-overlapping convex polygons in a set of n points in the plane is Θ(n √ k). This bound was already proved in the dual plane by Halperin and Sharir [HS92]. However, our proof is much simpler and uses the Crossing Lemma applied to the collection of edges of the given polygons. Similar results are obtained for more restricted collections of polygons. We then generalize these results to bound the total complexity of k distinct non-overlapping