Tel-Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science Combinatorial Problems in Computational Geometry Thesis submitted for the degree of “Doctor of Philosophy” by Shakhar Smorodinsky Under the supervision of Prof. Micha Sharir Submitted to the Senate of Tel-Aviv University June 2003 The work on this thesis was carried out under the supervision of Prof. Micha Sharir iii iv The thesis is dedicated to my parents, Meir and Nechama Smorodinsky, whom I love and who inspired me to love science. To my brother Rani and his family Tami, Guy, Adi, Omer. To Saba Eliahu (Niutek, who was recently upgraded to Niutekle), and to Savta Dora. v vi Acknowledgments I would like to express my deepest gratitude to my advisor Micha Sharir, who taught and inspired me, and spent so much time with me, discussing many problems (some of which are part of this thesis) and reading and commenting on this manuscript. More than anything, working with Micha was a profound pleasure. I would also like to thank J´anosPach who inspired me by his enthusiasm to count crossings in flat-land and who spent time with me discussing various geometric combi- natorial problems. I would also like to thank Pankaj Agarwal, Noga Alon, Boris Aronov, Alon Efrat, Guy Even, Zvika Lotker, J´anosPach, Sariel Har-Peled, Dana Ron for many helpful and stimulating discussions of scientific problems. I would like to thank some of my co-authors with whom I closely worked: Boris Aronov, Guy Even, Sariel Har-Peled, Zvika Lotker, J´anosPach, Rom Pinchasi, Micha Sharir. Finally, I would like to thank my friends from the dark open-space: Adi Avidor, Hadar Benayamini, Irit Dinur, Eti Ezra, Efi Fogel, Omer Friedland, Eran Halperin, Sariel Har-Peled, Guy Kindler, Zvika Lotker, Manor Mendel, Hayim Shaul, Oded Schwartz, with all of whom I discussed science, played chess or went for lunch together. vii viii Abstract In this thesis we study a variety of problems in combinatorial and computational ge- ometry, 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 Ap- plications: 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(m3=n2), 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 inter- section 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 IRd 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 S0 of S of size at least Ω(m2=n2) with nonempty intersection. This is asymptotically tight and improves the previously best known bound (see [CEG+94]). 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 complexityp 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 ix convex polytopes that are spanned by n points in IR3, where the complexity of a polytope is the total number of its facets. We show an upper bound of O(n2k1=3). This bound was already known in the dual space [AD01] but our proof is much simpler. The proof relies on a Crossing Lemma for triangles spanned by a finite point set in IR3. Additional bounds are obtained for more restricted classes of polytopes. In Chapter 4 we prove that the maximum number of k-sets in a set P of n points in IR3, i.e., the subsets P 0 of P of cardinality k for which there exists a halfspace H such that P 0 = P \ H, is O(nk3=2). This improves the previously best known bound of O(nk5=3) (see [DE94, AACS98]). The technique used to obtain this result is to establish a Crossing Lemma for triangles and points in IR3, which provides a lower bound on the number of crossing pairs of k-triangles, where a k-triangle is a triangle spanned by a triple of points of P such that the plane that contains this triangle passes above exactly k other points of P . Combining this with an upper bound implied by Lov´asz’Lemma then yields the asserted bound. The results of this chapter have been obtained with Micha Sharir and G´abor Tardos and appear in [SST01]. We also refer the reader to the recent book of Matouˇsek[Mat02] for a survey of the state of the art in the study of k-sets, including an exposition of the results of Chapter 4. In Chapter 5 we study a variety of problems involving certain types of extreme configurations in arrangements of (x-monotone) pseudo-lines, i.e., graphs of continuous totally-defined functions, each pair of which intersect in exactly one point. For example, we obtain a very simple proof of the bound O(nk1=3) on the maximum complexity of the k-th level in an arrangement of n pseudo-lines, which becomes even simpler in the case of lines. We thus simplify considerably previous proofs by Dey [Dey98] and by Tamaki and Tokuyama [TT97]. We also consider diamonds and anti-diamonds in (simple) pseudo-line arrangements, where a diamond is a pair u; v of vertices, so that u lies in the double wedge of v (consisting of all points lying above one curve that passes through v and below the other such curve) and vice versa, and an anti-diamond is a pair u; v where neither u nor v lies in the other double wedge. We show that the maximum size of a diamond-free set of vertices in an arrangement of n pseudo-lines is 3n ¡ 6, by showing that the induced graph (where each vertex of the arrangement is regarded as an edge connecting the two incident curves) is planar, simplifying considerably a previous proof of the same fact by Tamaki and Tokuyama [TT97]. Similarly, we show that the maximum size of an anti-diamond-free set of vertices in an arrangement of n pseudo- lines is 2n ¡ 2, improving a bound of 2n ¡ 1 due to Katchalski and Last [KL98] and reproducing a result independently obtained by Valtr [Val99]. We also obtain several additional results, which are listed in the introduction. The results of this chapter have been obtained with Micha Sharir and appear in [SS03a]. (ii) Conflict-Free Coloring of Points and Regions: Motivated by frequency assignment problems in cellular networks, we introduce and study in Chapter 6 new coloring problems of the following flavor: What is the minimum number f(n) such that one can assign colors to any set P of n points in the plane, using a total of at most f(n) colors, such that this coloring have the following property (which we refer to as Conflict-Free coloring or CF-coloring for short): For any disc d in the plane, with nonempty intersection with P , there is at least one point of P inside d which has a x unique color among the points of P \ d. We show that f(n) = O(log n), which is asymptotically tight in the worst case. We extend this result to many other classes of ranges (other than disks). A major tool in deriving these bounds is the introduction of a generalized variant of a Delaunay graph on P , whose edges connect pairs u; v 2 P if there exists a range of the type under consideration whose intersection with P is just the pair fu; vg. We show that the existence of large independent sets in this graph leads to a Conflict-Free coloring of P with a small number of colors. We also study the dual type of problems, where we want to color a given set R of ranges, so that for each point p there is a range in R with a unique color among the ranges of R that contain p.