Combinatorial Problems in Computational Geometry

Tel-Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science

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 JánosPach who inspired me by his enthusiasm to count crossings in flat-land and who spent time with me discussing various geometric combinatorial problems. I would also like to thank Pankaj Agarwal, Noga Alon, Boris Aronov, Alon Efrat, Guy Even, Zvika Lotker, JánosPach, 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, JánosPach, 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 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 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 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 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 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

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 Lovász’Lemma then yields the asserted bound. The results of this chapter have been obtained with Micha Sharir and Gábor 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 ∈ P if there exists a range of the type under consideration whose intersection with P is just the pair {u, v}. 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. For example, we show that any set of n pseudo-discs in the plane can be CF-colored using O(log n) colors. We show a strong relation between CF-coloring a finite set of ranges to the complexity of the union of any subset of these ranges. We thus generalize our result (on pseudo-disks) to any collection of n regions such that any subset of them has “low” union complexity. We also generalize this new notion of CF-coloring of regions to k-CF-coloring, where we require that for each point p there is a color that appears at least once but at most k times among the regions that contain p, for some fixed integer k. For example, we show that there exists a collection of n balls in IR3 for which in any CF-coloring, n colors are necessary, but one can k-CF- color any collection of n balls in IR3 with O(n1/k) colors. An analogous generalization to k-CF-coloring a range space (i.e., coloring a set of points with respect to a given collection of ranges) is studied. We show a relation between the k-CF-coloring problem of a range space, to its VC-dimension. Some of the the results of this chapter have been obtained with Guy Even, Zvika Lotker and Dana Ron in [ELRS03], and with Sariel Har-Peled in [HPS03].

xi xii Contents

Acknowledgments vii

Abstract ix

1 Introduction 1

2 Point-Selection Lemmas 9 2.1 Introduction ...... 9 2.2 Discs Spanned by Points in IR2 ...... 11 2.3 Pseudo-discs and Points in IR2 ...... 12 2.3.1 Yet Another Approach for Planar Selection Lemmas ...... 17 2.4 Balls and Points in Higher Dimensions ...... 19 2.4.1 Lines Stabbing Discs in IR3 ...... 24 2.5 Upper Bounds ...... 25 2.6 Axis-Parallel Rectangles ...... 26 2.6.1 An Upper Bound ...... 28 2.7 Open Problems ...... 31

3 Triangles and Points in IR3 33 3.1 Introduction ...... 33 3.2 Polygons in Point Sets and Concave Chains in Arrangements of Lines . 33 3.3 Triangles and Points in IR3 ...... 36 3.4 Polytopes Spanned by Point Sets ...... 38 3.4.1 Lower Bound ...... 39 3.4.2 Open Problems ...... 40

4 k-Sets in IR3 41 4.1 Introduction ...... 41 4.2 k-Sets and Triangles in IR3 ...... 41 4.2.1 An Overview of Our Technique ...... 41 4.2.2 Proof of the Theorem ...... 43 4.2.3 Proof of Theorem 3.7 ...... 48 4.2.4 Open Problems ...... 49

xiii 5 Generalized Geometric Graphs and Pseudo-line Arrangements 51 5.1 Introduction ...... 51 5.2 Drawing Pseudo-line Graphs ...... 56 5.3 The Complexity of a k-Level in Pseudo-line Arrangements ...... 59 5.4 Yet Another Proof for Incidences and Many Faces in Pseudo-line Ar- rangements ...... 61 5.5 Graphs in Pseudo-line Arrangements without Anti-Diamonds ...... 62 5.6 Pseudo-line and Thrackles ...... 64

6 Conflict-Free Coloring Problems 67 6.1 Introduction ...... 67 6.2 A General Framework ...... 70 6.3 CF-Coloring of Range Spaces ...... 72 6.3.1 Coloring Points in the Plane with Respect to Discs ...... 72 6.3.2 Coloring Points in IR3 with Respect to Halfspaces ...... 73 6.3.3 Axis-parallel Rectangles ...... 74 6.4 CF-Coloring of Regions ...... 77 6.4.1 From Discs to Halfspaces ...... 77 6.4.2 CF-Coloring of Regions with Low Union Complexity ...... 78 6.5 CF-Coloring of Simple Geometric Regions in the Plane ...... 82 6.5.1 Conflict-Free Coloring of Axis-Parallel Rectangles ...... 82 6.5.2 CF-Coloring of “Well Behaved” Unbounded Regions ...... 83 6.6 Miscellaneous CF-Coloring Problems ...... 85 6.6.1 CF-Coloring of Points with respect to Balls and Halfspaces in Higher Dimensions ...... 85 6.6.2 CF-Coloring of Half-Slabs ...... 85 6.7 Relaxing the Notion of Conflict-Free Coloring ...... 86 6.7.1 k-CF-Coloring of a Range-space ...... 86 6.7.2 k-CF-Coloring of Range Spaces with Finite VC-Dimension . . . 87 6.7.3 k-CF-Coloring of Regions ...... 88

Bibliography 92

xiv Chapter 1

Introduction

Selection lemmas. In Chapter 2 we study several point selection problems of the following flavor. Let P be a set of n points in IRd, and let D be a family of m distinct objects of some fixed kind (such as balls, discs, triangles, etc.), so that the boundary of each object in D passes through some distinct tuple of points of P . We wish to assert that there always exists a point that is contained in many objects of D, or that there exists a line that stabs many objects of D, etc. Problems of this kind have been studied in the past, see the recent book of Matouˇsek d [Mat02]. Bárány [Bár82]has shown that for any¡ finite¢ set P of n points in IR there n d+1 is always a point that lies in the interior of Ω( d+1 ) = Ω(n ) simplices spanned by P , that is, simplices whose vertices belong to P (see also [BF84]). In other words, a fixed percentage of all the simplices spanned by P have a nonempty intersection which is asymptotically tight. In the plane, this means that for any set P of n points, there exists a point that lies in the interior of Ω(n3) triangles with vertices from P . This raises the following more general question: For given integer parameters n and t, what is the maximum number f(n, t), such that, for any set P of n points in IR2 and any set T of t triangles spanned by P , there exists a point that lies in the interior of at least f(n, t) triangles of T ? Aronov et al. [ACE+91] have shown that f(n, t) = Ω(t3/(n6 log5 n)). Their motivation was to derive an upper bound on the number of halving planes of a finite set of points in IR3 (i.e., planes that pass through a triple of the given points, and partition the remaining points into two subsets of equal size, for n odd). Indeed, using the above bound, combined with Lovász’Lemma [Lov71] for halving triangles (i.e., the triangles spanned by the triples of points that span the halving planes), Aronov et al. were able to show that any set of n points in IR3 determines at most O(n8/3 log5/3 n) halving planes. This result is improved in this thesis to O(n5/2). See Chapter 4 for the improvement, as well as for more details concerning Lovász’Lemma and its applications. A different motivation for this type of problems was given by Chazelle et al. [CEG+94]. 2 Introduction

Their goal was to reduce the size of Delaunay triangulations for finite point sets in IR3. For such a set P , the Delaunay triangulation, D(P ), consists of all tetrahedra spanned by the points of P whose circumscribed spheres enclose no point of P in their interior (see, e.g., [dBvKOS00]). Depending on how the points are distributed, the number of tetrahedra can vary between linear and quadratic in n. The goal in [CEG+94] was to find, for any set P on n points in IR3, an additional set Q of a small number of points such that D(P ∪ Q) is guaranteed to have only a small number of tetrahedra. The approach in [CEG+94] was to find a point q that lies inside “many” spheres circumscribing the tetrahedra of the original Delaunay triangulation. Adding q to P would remove all corresponding tetrahedra from D(P ) and replace them by at most a linear number of new tetrahedra, all incident to q. Thus, the problem of slimming down 3-dimensional Delaunay triangulations can be attacked by showing that if there are “many” circumscribing spheres then there must be a point enclosed by “many” of them. The main tool used in [CEG+94] was the following d-dimensional selection lemma for axis-parallel boxes: For any set P of n points in IRd and any set of m distinct d-dimensional boxes, each of which is axis-parallel and determined by a unique pair of points of P (as opposite vertices), there is a point that is covered by Ω(m2/(n2 log2d−2 n)) of the boxes. Then, observing that any diametrical ball spanned by two points p and q (i.e., the ball for which pq is a diameter) must contain the box determined by p and q, it follows that the same lower bound also holds for points covered by diametrical balls. Using additional arguments, the analysis was extended to any collection of m balls, each having a bounding sphere passing through a distinct pair of points of P , showing that there always exists a point enclosed by at least Ω(m2/(n2 log2d n)) of the balls (a slightly weaker bound than that for diametrical balls). The problem that motivated the study of [CEG+94], namely slimming Delaunay triangulations in 3-space, has since been further improved using a totally different approach (see [BEG94]). As already mentioned, we improve in Chapter 4 the bound on the other problem that motivated the study of [ACE+91], namely the problem of halving planes. Nevertheless, point selection theorems of this kind remain of independent interest. In particular, the bounds obtained in [ACE+91, CEG+94] are not shown to be optimal (and, as shown in Chapter 2, many of them are not optimal).

The Crossing Lemma and its applications. Let G = (V,E) be a simple graph. A drawing of G in the plane is a mapping that maps each vertex v ∈ V to a point in the plane, and each edge e = uv of E to a Jordan arc connecting the images of u and v, such that no three arcs are concurrent at their relative interiors, and the relative interior of no arc is incident to a vertex.

Theorem 1.1. (The Crossing Lemma) Let G = (V,E) be a simple graph with 3 |E| ≥ 4 |V |. Then, in any drawing of G in the plane there must be at least |E| pairs 64|V |2 of crossing edges. The proof of this theorem is due to Leighton [Lei83] and Ajtai et al. [ACNS82] (see also [AS92, PA95, Sha03]). A beautiful probabilistic version of this proof was Introduction 3

classified as a proof from THE BOOK (in Erd˝os’terminology), and is presented in [AS92, AZ98].1 A surprising relation between the Crossing Lemma and other geometric combinatorial problems, such as incidences between points and lines, was discovered by Székely [Szé97].As we demonstrate numerous times in the first part of the thesis, the probabilistic technique used in that proof is very powerful and general. In Chapter 2 we use a similar probabilistic analysis to obtain bounds for point- selection theorems involving points and balls in IRd, and points and pseudo-discs in the plane. We also use the Crossing Lemma to obtain bounds for point-selection theorems involving points and other regions in IR2 with “low” union complexity. Here is a sample of a few of the results that we obtain:

1. 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 bound is asymptotically tight.

2. For any set P of n points in the plane and any set C of m ≥ 4n distinct disks, each bounded by a circle passing through a distinct triple of points of P , there exists a point p ∈ P that is covered by Ω(m3/2/n3/2) discs of C. This bound is asymptotically tight.

3. For any set P of n points in the plane and any set R of m ≥ 8n log n axis-parallel rectangles, each having two points of P as two opposite vertices, there exists a m2 subset of R of size at least with nonempty intersection. (A diﬀerent 512n2 log2 n proof of this result is given in [CEG+94].)

The results of Chapter 2 have been obtained with Micha Sharir and appear in [SS03b].

Polytopes spanned by points. In Chapter 3 we use the Crossing Lemma to obtain simple and elegant bounds for the total complexity of k concave chains in an arrangement of n lines (or pseudo-lines) in the plane. Our proof is in the dual plane, where we bound the total complexity of k distinct non-overlapping convex polygons that are spanned by a ﬁnite set P of n points (namely, each such polygon is the convex hull 0 of some subset P ⊂ P ). Speciﬁcally, we show that the maximum total complexity√ of k distinct non-overlapping polygons that are spanned by n points is Θ(n k). If, in addition, we assume that any line l can be tangent to at most one such polygon at a point p ∈ P , then the maximum total complexity of these polygons is Θ(nk1/3) for k ≤ n and Θ(n2/3k2/3) for k ≥ n. Next, we generalize the Crossing Lemma to three dimensions as follows: We say 3 that two triangles ∆1, ∆2 in IR , with a common vertex, cross, if their relative interiors cross; see Figure 3.2. Note that if ∆1 = abc and ∆2 = ade cross, then either the straight line segment de crosses the relative interior of ∆1 or the straight line segment bc crosses

1The constant 64 has recently been improved; see [PT97]. 4 Introduction

2 the relative interior of ∆2. We show that in any set of t > 2n triangles spanned by n points in IR3 in general position, there must be at least Ω(t3/n4) crossing pairs of triangles (with a common vertex). This enables us to prove that for any set of n points in IR3 and any set of t > 2n2 triangles spanned by those points, there exists a line that stabs Ω(t3/n6) of the given triangles. This fact is already known (see [DE94]), but our proof is much shorter and simpler. This latter fact, combined with the aforementioned Lov´asz’Lemma for halving triangles, provides an O(n8/3) upper bound on the number of halving triangles in a set of n points in IR3. In addition, using this three-dimensional analog of the Crossing Lemma, we provide a non-trivial upper bound on the total complexity of a set of k distinct non-overlapping convex polytopes spanned by n points in IR3. Speciﬁcally, we show that the total complexity k such polytopes is O(n2k1/3). This bound is already known (in its dual version) [AD01] but our proof is is much shorter and simpler. If, in addition, we assume that any plane π is tangent to at most one such polytope at any given edge pq, then the maximum total complexity of the these polytopes is O(n3/2k1/2).

k-Sets. Let S be a set of n points in IRd in general position. A k-set of S is a subset S0 ⊂ S such that S0 = S ∩ H for some halfspace H and |S0| = k. The study of k-sets has a long and rich history, and many applications, both in discrete and in computational geometry. The problem of determining tight asymptotic bounds on the maximum number of k-sets in an n-element set is one of the most intriguing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [CSY87, EW85], the problem has caught the attention of computational geometers as well [ACE+91, DE94, EVW97, Sha91, Wel86]. A close to optimal solution

for the problem remains elusive even in the plane. The best asymptotic√ upper and lower bounds in the plane are O(nk1/3) (see [Dey98]) and n · 2Ω( log k) (see [Tót01]), 3 5/3 respectively. The best known upper√ bound in IR is O(nk ) [DE94], and the best known lower bound is Ω(nk2Ω( log k)) [Tót01]. In Chapter 4 we obtain an improved upper bound of O(nk3/2) on the number of k-sets in a set of n points in IR3. The main result that enables us to obtain the improved bound is a lower bound on the number of pairs of crossing pairs of halving triangles.2 A careful inspection of the halving triangles in a set of n points reveals that it possesses an important property, which we refer to as the antipodality property; see [SST01] and Chapter 4. We then show that if T is a set of t À n2 triangles spanned by n points in IR3 that has the antipodality property, then the number of crossing pairs of triangles (with a common vertex) is Ω(t2/n), improving significantly the general lower bound Ω(t3/n4) mentioned above. This implies that there is a line that stabs Ω(t2/n3) of the given triangles. Again, combined with Lovász’ Lemma for halving triangles, this provides the desired bound t = O(n5/2). The results of Chapter 4 are joint work with Micha Sharir and Gábor 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.

2It is indeed suﬃcient to establish this bound for the case k = (n − 3)/2 of halving triangles. Introduction 5

Geometric graphs and their generalizations. In Chapter 5 we study various combinatoric problems related to arrangements of pseudo-lines. A set Γ of n pseudo-lines in the plane is a collection of graphs of n continuous totally-deﬁned functions, each pair of which intersect in exactly one point, so that the curves cross each other at that point. Let Γ be a collection of n pseudo-lines in general position (i.e., no three curves in Γ pass through a common point) and let E be a subset of the vertices of the arrangement A(Γ). E induces a graph G = (Γ,E) on Γ (in what follows, we refer to such a graph as a pseudo-line graph). For each pair (γ, γ0) of distinct pseudo-lines in Γ, we denote by W (γ, γ0) the double wedge formed between γ and γ0, that is, the (open) region consisting of all points that lie above one of these pseudo-lines and below the other. We also denote by W c(γ, γ0) the complementary (open) double wedge, consisting of all points that lie either above both curves or below both curves.

Deﬁnition 1.2. We say that two edges (γ, γ0) and (δ, δ0) of G form a diamond if the point γ ∩ γ0 is contained in the double wedge W (δ, δ0), and the point δ ∩ δ0 is contained in the double wedge W (γ, γ0). See Figure 5.1(i).

Deﬁnition 1.3. We say that two edges (γ, γ0) and (δ, δ0) of G form an anti-diamond if the point γ ∩ γ0 is not contained in the double wedge W (δ, δ0), and the point δ ∩ δ0 is not contained in the double wedge W (γ, γ0); that is, γ ∩ γ0 lies in W c(δ, δ0) and δ ∩ δ0 lies in W c(γ, γ0). See Figure 5.1(ii).

Deﬁnition 1.4. (a) A collection S of x-monotone bounded Jordan arcs is called a collection of pseudo-segments if each pair of arcs of S intersect in at most one point, where they cross each other. (b) S is called a collection of extendible pseudo-segments if there exists a set Γ of pseudo-lines, such that each s ∈ S is contained in a unique pseudo-line of Γ.

See [Cha03] for more details concerning extendible pseudo-segments. Note that not every collection of pseudo-segments is extendible, as shown by the simple example depicted in Figure 5.2.

Deﬁnition 1.5. Consider a drawing of a graph G = (V,E) in the plane. If the images of the edges of E form a family of extendible pseudo-segments then we refer to the drawing of G as an (x-monotone) generalized geometric graph.

(The term geometric graphs is usually reserved to drawings of graphs where the edges are drawn as straight segments. Drawing the edges without any restriction yields so-called topological graphs.) The k-level in an arrangement of a set Γ of n pseudo-lines is the (closure of) the set of all points that lie on curves of Γ and have exactly k other curves passing below them. When Γ consists of straight lines, the number of vertices of the k-level of the arrangement of the lines in Γ is proportional to the number of k-sets in the set Γ∗ of dual points of the lines in Γ. Thus k-levels in arrangements of pseudo-lines can be viewed as a generalization of planar k-sets. 6 Introduction

In Chapter 5 we establish an equivalence between pseudo-line graphs and generalized geometric graphs. As an immediate corollary we show that if a pseudo-line graph G = (Γ,E) is diamond-free, then |E| ≤ 3n − 6. This fact has been proven by Tamaki and Tokuyama [TT97], using a considerably more involved and complicated argument. This was the underlying theorem that enabled them to extend Dey’s improved bound of O(nk1/3) on the complexity of the k-level in an arrangement of lines [Dey98], to arrangements of pseudo-lines. Note that the planarity of G is obvious for the case of lines: If we dualize the given lines into points, using the duality y = ax + b 7→ (a, b) and (c, d) 7→ y = −cx + d, presented in [Ede87], and map each edge (γ, γ0) of G to the straight segment connecting the points dual to γ and γ0, we obtain a crossing-free drawing of G. Hence, the above fact is a natural (though harder to derive) extension of this property to the case of pseudo-lines. In addition to the simpliﬁed proof of Tamaki and Tokuyama’s bound, we tackle explicitly the k-level problem, and provide a new and very simple proof of the bound O(nk1/3), which applies to both cases of lines and pseudo-lines.

Deﬁnition 1.6. A thrackle is a drawing of a graph in the plane so that every pair of edges either have a common endpoint and are otherwise disjoint, or else they intersect in exactly one point where they cross each other.

The notion of a thrackle is due to Conway, who conjectured that the number of edges in a thrackle is at most the number of vertices. The study of thrackles has drawn much attention. Two recent papers [LPS97] and [CN00] obtain linear bounds for the size of a general thrackle, but with constants of proportionality that are greater than 1. The conjecture is known to hold for straight-edge thrackles [Pac99], and, in Section 5.6, we extend the result, and the proof, to the case of thrackles drawn as generalized geometric graphs. We also study pseudo-line graphs G = (Γ,E) that do not have any anti-diamond. We show that in this case |E| ≤ 2n − 2. This fact is an extension, to the case of pseudo-lines, of a (dual version of a) theorem of Katchalski and Last [KL98], reﬁned by Valtr [Val98b], both solving a problem posed by Kupitz. The theorem states that a straight-edge graph on n points in the plane, which does not have any pair of parallel edges, has at most 2n − 2 edges. A pair of segments e, e0 is said to be parallel (or avoiding) if the line containing e does not cross e0 and the line containing e0 does not cross e. (For straight edges, this is equivalent to the condition that e and e0 are in convex position.) The dual version of a pair of parallel edges is a pair of vertices in a line arrangement that form an anti-diamond. Hence, our result on anti-diamond-free graphs is indeed an extension of the result of [KL98, Val98b] to the case of pseudo-lines. (A similar simpliﬁed proof has been independently obtained by Valtr [Val99].) Finally, using the Crossing Lemma we provide yet another simple proof of the following well-known result.

Theorem 1.7. (a) The maximum number of incidences between m distinct points and n distinct pseudo-lines is Θ(m2/3n2/3 + m + n). Introduction 7

(b) The maximum number of edges bounding m distinct faces in an arrangement of n pseudo-lines is Θ(m2/3n2/3 + n). Theorem 1.7 was originally obtained by Clarkson et al. [CEG+90], extending the original result of Szemerédiand Trotter [ST83]. Our proofs are in some sense ‘dual’ to the proofs based on Székely’s technique [DP98, Szé97]. The proof of case (b) can be extended to yield the following result, recently obtained in [AAS03], where it has been proved using the dual approach, based on Székely’s technique. Theorem 1.8. The maximum number of edges bounding m distinct faces in an arrangement of n extendible pseudo-segments is Θ((m + n)2/3n2/3). The results of Chapter 5 have been obtained with Micha Sharir and appear in [SS03a].

Conflict-free coloring problems. In Chapter 6 we study coloring problems that arise in frequency assignment to cellular antennas. Specifically, cellular networks are heterogeneous networks with two different types of nodes: base stations (that act as servers) and clients. The base stations are interconnected by an external fixed backbone network. Clients are connected only to base stations; connections between clients and base stations are implemented by radio links. Fixed frequencies are assigned to base stations to enable links to clients. Clients, on the other hand, continuously scan frequencies in search of a base station with good reception. The fundamental problem of frequency assignment in cellular networks is to assign frequencies to base stations so that every client, wherever s/he is, can be served by some base station, in the sense that the client is located within the range of the station and no other station within its reception range has the same frequency. The goal is to minimize the number of assigned frequencies since the frequency spectrum is limited and costly. In abstract setting, the problem can be formulated as follows: CF-coloring of regions: Given a finite family S of n regions of some fixed type (such as discs, pseudo-discs, axis-parallel rectangles, etc.), what is the minimum integer k, such that one can assign a color to each region of S, using a total of at most k colors, such that the resulting coloring has the following property: For each point p ∈ ∪b∈S b there is at least one region b ∈ S that contains p in its interior, whose color is unique among all regions in S that contain p in their interior (in this case we say that p is being ‘served’ by that color). We refer to such a coloring as a Conflict-Free coloring of S (CF-coloring in short). Suppose we are given a set of n base stations, also referred to as antennas. Assume, for simplicity, that the area covered by a single antenna is a disc in the plane. Namely, the location of each antenna (base station) and its radius of transmission are fixed and known (the transmission radii of the antennas are not necessarily equal). In Chapter 6 we show that in this case, one can find an assignment of frequencies to the antennas with a total of at most O(log n) frequencies, such that each antenna is assigned exactly one of the frequencies and the resulting assignment is conflict-free in the above sense. Furthermore, we show that this bound is worst-case optimal. 8 Introduction

Thus, we show that any family of n discs in the plane has a CF-coloring with O(log n) colors and that this bound is tight in the worst case. Furthermore, such a coloring can be found in polynomial time. We also study other variants of this problem. For example we study CF-coloring problems of different types of regions (not necessarily discs).3 We show a strong relation between CF-coloring of a finite set of regions R and the complexity of the union of the regions of R. For example we show that if R is a set of n Jordan regions (not necessarily convex) with the property that any m regions of R have union complexity O(m) (e.g., pseudo-discs have this property; see [KLPS86]), then R can be CF-colored with a total of O(log n) colors. We also study a generalization of that problem to what we call k-CF-coloring. Instead of requiring that a point is ‘served’ if there is a region with a ‘unique’ color that contains the point, we require that there is a color that appears (at least once and) at most k times in the set of regions containing that point. This generalization seems to be very natural and provides important insights into this type of problems. Indeed, for example, we show 3 that for any n there exists a collection√ R of n balls in IR such that n colors are needed in any CF-coloring of R but O( n) colors suffice for 2-CF-coloring of R, and in general O(n1/k) colors suffice for k-CF-coloring of R. We also study CF-coloring problems for range spaces and show a relation between this notion to that of CF-coloring of regions. CF-coloring of a range space: A set P of n points in IRd and a set R of ranges (for example, the set of all discs in the plane) form what is called a range space (P, R) (see [PA95]). We seek the minimum integer k, such that one can color the points of P by k colors, so that for any r ∈ R with P ∩ r 6= ∅, there is at least one point q ∈ P ∩ r that is assigned a unique color among all colors assigned to points of P ∩ r (in this case we say that r is ‘served’ by that color). We refer to such a coloring as a Conflict-Free coloring of (P, R) (CF-coloring in short). We derive various bounds for CF-coloring of range spaces. For example, we show:

1. Any set of n points in the plane can be CF-colored with O(log n) colors with respect to discs, and this bound is asymptotically tight.

2. Any set of n points in IR3 can be CF-colored with O(log n) colors with respect to halfspaces. This bound is asymptotically tight.

3. Any set of n points in IRd can be CF-colored with O(n1−1/2d−1 ) colors, with respect to axis-parallel boxes.

An analogous notion to that of k-CF-coloring of regions can also be deﬁned and studied in the context of a range space. We also show a strong relation between k-CF- coloring of a range space and its VC-dimension [PA95]. The results of Chapter 6 appear in two papers. The ﬁrst paper is a joint work with Guy Even, Zvi Lotker and Dana Ron [ELRS03]. The second paper is joint work with Sariel Har-Peled [HPS03].

3As a matter of fact, antennas are generally directional, so the region controlled by an antenna is a circular sector rather than a full disc. Chapter 2

Point-Selection Lemmas

2.1 Introduction

In this chapter we study several point selection problems of the following kind: We are given a set P of n points in IRd and a collection D of m distinct objects of some simple shape, so that the boundary of each object of D passes through some distinct tuple of points in P , and we wish to assert that there always exists a point (which, in some versions, is required to belong to P while in other versions can be arbitrary) that lies in many objects of D. We improve (and in many cases tighten) and generalize some of the bounds obtained in [CEG+94], e.g., for the cases where the objects in D are discs or balls, using a fairly simple and more direct approach to tackle the problem, using a generalization of the probabilistic proof technique of the Crossing Lemma mentioned in the introduction. We outline the main ideas employed in all of our results, using the following specific problem: Given a set P of n points and a set C of m distinct discs in the plane, where the boundary of each disc passes through a distinct pair of points of P , we wish to show that there is a point in P that lies in “many” of the given discs. To do so, we first define a configuration to be a pair of a point in P and a disc in C, such that the point lies inside the disc. We aim to show that there are many such configurations. Using standard properties of the Delaunay triangulation of P , we show that if m is large enough (specifically, larger than 3n), then there exists at least one configuration. Then, using a random sampling technique, similar to that used in the proof of the Crossing Lemma of Leighton and Ajtai et al. (see [AZ98, PA95, Sha03, Mat02]), we derive a lower bound f(n, m) on the number of such configurations. Finally, by the pigeonhole principle, at least one of the points of P participates in at least f(n, m)/n configurations, yielding the desired lower bound for point selection. As just mentioned, the technique used in all our proofs can be viewed as an extension of the probabilistic proof technique of the Crossing Lemma. Even though this technique is rather well known, there are only a few applications of it in geometric settings. Our results illustrate the versatility of the technique and enrich the set of situations in which it can be applied. This will also be demonstrated in Chapters 3–5. We now summarize the main results and present the outline of this chapter. In 10 Point-Selection Lemmas

Table 2.1: Summary of point selection bounds. objects/spanned by dim prev. bound new bound stab. pt in P m2 m2 discs/point pairs 2 Ω( n2 log4 n ) Ω( n2 ) yes m3/2 discs/triples of points 2 - Ω( n3/2 ) yes m2 pseudo-discs/point pairs 2 - Ω( n2 ) yes m3/2 pseudo-discs/triples of points 2 - Ω( n3/2 ) yes balls/point pairs d Ω( m2 ) Ω( m2 ) no n2 log2d(m2/n) n2 m2 lines stabbing discs/point pairs 3 - Ω( n2 ) - m2 m2 axis-parallel rectangles 2 Ω( n2 log2 n ) O( n2 log(n2/m) ) no

Section 2.2 we introduce our technique, by showing that, for any set P of n points in the plane and any set of m distinct discs, each of which is spanned by (i.e., its boundary passes through) a distinct pair of points (resp., a triple of points) of P , there is a point in P that lies in Ω(m2/n2) (resp., Ω(m3/2/n3/2)) discs. A simple application of the latter bound is an alternative derivation of the bound O(nk2) on the overall complexity of the j-order Voronoi diagrams of a set P of n points in the plane, for j = 1, . . . , k (see [Ede87]). We describe this application in Section 2.2. In Section 2.3, we show how to generalize these results to arbitrary families of pseudo-discs (regions bounded by closed Jordan curves, every two of which intersect at most twice). Section 2.4 deals with the higher dimensional analog of this problem, involving n points and m distinct balls in IRd spanned by distinct pairs of points of P . We show that there exists a point (not necessarily of P ) that lies inside Ω(m2/n2) balls. We also study a variant where we have n points in IR3 and m distinct discs, each spanned by a distinct pair of points. We show that there exists a line that stabs Ω(m2/n2) of the given discs. In Section 2.5 we show that all the results mentioned so far are asymptotically tight in the worst case. In Section 2.6 we show that for any set P of n points in the plane and any set of m distinct axis-parallel rectangles, each of which contains a pair of points of P as opposite vertices, there exists a point (not necessarily of P ) that lies inside Ω(m2/n2 log2 n) rectangles. This bound was proved in [CEG+94], but the proof technique that we present is totally diﬀerent (and follows the same general approach used in the preceding sections). We also present an improved upper bound. Namely, for any n and m we construct a set P of n points in the plane and m axis-parallel rectangles spanned by pairs of points of P such that no point in the plane lies inside more than O(m2/(n2 log(n2/m))) rectangles. With the exception of axis-parallel rectangles, each of our results either improves (and tightens) the previous corresponding result of [CEG+94], or is the ﬁrst nontrivial bound for the problem. Furthermore, the two-dimensional results of Sections 2.2 and 2.3 are stronger than that of [CEG+94] in the additional sense that they guarantee the existence of a stabbing point that belongs to P , rather than an arbitrary point in the plane. We also make progress on the case of axis-parallel rectangles, by providing the aforementioned upper bound, since no sub-(m2/n2) bound has been previously known. Table 2.1 summarizes the results obtained in this chapter. The results of this chapter are joint work with Micha Sharir and appear in [SS03b]. 2.2 Discs Spanned by Points in IR2 11

2.2 Discs Spanned by Points in IR2

Theorem 2.1. Let P be a set of n points and let D be a set of m ≥ 4n distinct discs in IR2. (i) If the boundary of each disc passes through a pair of points of P and for any pair of points p, q ∈ P there is at most one disc in D whose boundary passes through p and q, then there exists a point of P that is covered by Ω(m2/n2) discs. (ii) If the boundary of each disc passes through a triple of points of P and for any triple of points p, q, r ∈ P there is at most one disc in D whose boundary pases through p, q, r, then there exists a point of P that is covered by Ω(m3/2/n3/2) discs. Both bounds are tight in the worst case, in the strong sense that there are constructions involving n points and m discs, for which no point in the plane (not just points of P ) is covered by more than O(m2/n2) discs in case (i), or O(m3/2/n3/2) discs in case (ii).

First, we prove the following ‘bootstrapping’ lemma. Define a configuration to be a pair (p, d) ∈ P × D such that p lies in d, and p is not one of the two points (in case (i)) or three points (in case (ii)) that define (i.e., span) d.

Lemma 2.2. Let P and D be as in Theorem 2.1 and let X denote the number of conﬁgurations in P × D. Then X ≥ m − 3n in case (i), and X ≥ m − 2n in case (ii).

Proof. Suppose first that the points of P are in general position, in the sense that no four of them are co-circular. It is well known (see, e.g., [dBvKOS00]) that the number of pairs of points p, q ∈ P , such that there is an empty disc whose boundary passes through p and q (i.e., the interior of the disc contains no points of P ), is at most 3n − 6 (those pairs are the Delaunay edges of P ), and the number of triples of points p, q, r ∈ P such that the disc passing through them is empty, is at most 2n − 4 (those triples form the Delaunay triangles of P ). If the points are not in general position, the following modified property holds: The number of distinct pairs of points p, q ∈ P for which there is a disc whose boundary passes through p and q and which contains no other point of P in its closure is at most 3n − 6. Similarly, the number of distinct empty discs that pass through triples of points of P is at most 2n − 4. We present the proof of the first inequality, which proceeds by induction on m − 3n. For m − 3n ≤ 0 the claim is trivial. Assume that the claim holds for some non-negative integer k (namely, for m and n satisfying m − 3n = k). Suppose that m − 3n = k + 1. Since m > 3n, there must exist a nonempty disc d ∈ D, which generates at least one configuration with the points of P . After removing d from D we are left with m−1 discs, n points, and X0 configurations, where X ≥ X0 + 1. We have m − 1 − 3n = k, so we can apply the induction hypothesis to obtain X0 ≥ m−1−3n. Thus X ≥ X0 +1 ≥ m−3n. This completes the proof of the first claim of the Lemma. The proof of the second claim is similar. Proof of Theorem 2.1: Let X denote the number of configurations, as in Lemma 2.2. We aim to show that the number of such configurations is “large”. We take a random sample P 0 of the points in P by choosing each point independently with some fixed 12 Point-Selection Lemmas

probability p (to be determined later on). Let D0 denote the subset of discs in D, all of whose defining points are in P 0. Put n0 = |P 0|; m0 = |D0|, and let X0 denote the number of configurations all of whose defining points are in P 0. Consider first case (i) of the theorem. By Lemma 2.2 we have X0 ≥ m0 − 3n0. Note that X0, m0 and n0 are random variables, so the above inequality holds for their expectations as well. Hence (using linearity of expectation), E[X0] ≥ E[m0] − 3 E[n0]. It is easily seen that E[n0] = pn. We have E[m0] = p2m and E[X0] = p3X. Indeed, the probability that a given disc d ∈ D belongs to D0 is the probability that the two points defining d are chosen in P 0, which is p2 for any fixed d ∈ D. Similarly, the probability that a configuration of a point p ∈ P that is covered by a disc whose boundary passes through two other points r, q ∈ P is counted in X0 is p3. Substituting these values in the above inequality, we 3 2 m 3n get p X ≥ p m − 3pn, or X ≥ p − p2 . This inequality holds for any 0 < p ≤ 1, and m2 we choose p = 4n/m (by assumption, p ≤ 1) to obtain X ≥ 16n . By the pigeonhole X m2 principle, one of the points in P is covered by at least n ≥ 16n2 discs. This completes the proof of case (i) of the theorem. For case (ii), we have X0 ≥ m0 − 2n0, E[m0] = p3m, and E[X0] = p4X, which implies that p4X ≥ p3m − 2pn, or X ≥ m − 2n . This inequality holds for any 0 < p ≤ 1, and p p p3 m3/2 we choose p = 2 n/m (again, p ≤ 1), to obtain X ≥ 4n1/2 . As above, one of the points X m3/2 in P is covered by at least n ≥ 4n3/2 of the discs. This completes the proof of case (ii) of the theorem. The proofs of the worst-case optimality of these bounds are delegated to Section 2.5. 2 Remark: A simple application of the above analysis is an alternative derivation of the bound O(nk2) on the overall complexity of the first j-order Voronoi diagrams of a set P of n points in the plane, for j = 1, . . . , k (see [Ede87]). Specifically, the vertices of those diagrams are exactly the centers of discs whose boundaries pass through three points of P and containing at most k − 1 points of P in their interior. Let m denote the number of such discs. By the proof of Theorem 2.1, the number of configurations of a point in P inside such a disc is Ω(m3/2/n1/2). On the other hand, the number of such configurations is at most mk, since no disc contains more than k points in its interior. Solving the resulting inequality, we obtain m = O(nk2). (This proof is different from the one that uses Clarkson’s theorem on levels (see [Mat02]), which can also be used to achieve this bound.) Many other variants can also be tackled using the above analysis. For example, the maximum number of discs, each of whose boundary passes through a triple of points of P , so that no point of P is contained in more than k of them, is O(nk2/3), which is obtained using the upper bound nk on the number of such configurations. See also [Sha03] for related work.

2.3 Pseudo-discs and Points in IR2

In this section we generalize Theorem 2.1 to an arbitrary collection of pseudo-discs. We begin with several technical deﬁnitions and lemmas: Deﬁnition 2.3. A simple closed Jordan curve (resp., a simple Jordan arc) is the image of a continuous 1-1 mapping from the unit circle (resp., from [0, 1]) to IR2. 2.3 Pseudo-discs and Points in IR2 13

We next state the famous Jordan theorem for closed Jordan curves (see, e.g., [Moi77]): Theorem 2.4 (Jordan Theorem). Let γ be a simple closed Jordan curve. Then the complement of γ (i.e., IR2 \ γ) consists of exactly two connected components, one of which is bounded and the other is unbounded. For a simple closed Jordan curve γ, we say that a point p lies in the interior (resp., exterior) of γ if p lies in the bounded (resp., unbounded) connected component of the complement of γ. Lemma 2.5. Let γ be a simple closed Jordan curve and let p and q be two points in IR2 \γ. Then p and q lie in diﬀerent connected components of IR2 \γ if and only if every simple Jordan arc between p and q that intersects γ only at points where the curves cross each other, meets γ an odd number of times. See Figure 2.1(i).

Lemma 2.6. Let p, q be two points in the plane and let γ1, γ2, γ3 be three pairwise openly disjoint simple Jordan arcs with end-points p and q. Then the relative interior of exactly one of the arcs, say γ1, lies fully in the interior of the closed Jordan curve γ2 ∪ γ3. See Figure 2.1(ii).

p γ2 q p q γ1

γ3 γ

(ii) (i)

Figure 2.1: (i) A Jordan arc between p and q (shown dashed) intersects a closed Jordan curve, that separates p and q, an odd number of times. (ii) The relative interior of the arc γ1 is contained in the interior of the closed Jordan curve γ2 ∪ γ3.

The above two lemmas are easy consequences of the Jordan theorem. Deﬁnition 2.7. A family of closed Jordan regions is called a family of pseudo-discs if the boundaries of any two of its members are either disjoint or cross in exactly two points. Lemma 2.8. Let P be a set of n points in the plane. Let C be a family of m distinct pseudo-discs, such that every member of C has its boundary passing through a distinct pair of points p, q ∈ P , and such that all regions in C are empty (i.e., no point of P lies in the interior of any region in C). Then m ≤ 3n − 6. 14 Point-Selection Lemmas

Proof. Let G be the graph whose vertices are the points in P and whose edges are the m point pairs that deﬁne the regions of C. For an edge (p, q) of G, let cpq be the boundary curve of the region in C that passes through p and q. We embed G in the plane, so that the edge (p, q) is drawn along one of the two possible portions of cpq delimited by p and q, which we choose arbitrarily and denote it by γpq. We will show that in the above drawing of G, any two edges on four distinct vertices intersect an even number of times. This, combined with the Hanani-Tutte’s theorem [Tut70] (see also [Han34, LPS97]), implies that G is planar (and simple) and hence m ≤ 3n − 6. Assume to the contrary that there are four vertices of G, p1, q1, p2, q2, such that the

arc γp1q1 (γ1 for short) and the arc γp2q2 (γ2 for short) intersect an odd number of times. Since C is a family of pseudo-discs, any two such edges intersect at most twice. Hence if γ1 and γ2 intersect an odd number of times then they intersect exactly once. See Figure 2.2 for an illustration. Let c1 (resp., c2) denote the pseudo-disc whose boundary passes through p1 and q1 (resp., by p2 and q2). If γ2 intersects c1 exactly once, then, 2 by Lemma 2.5, p2 and q2 must lie in diﬀerent connected components of IR \ c1. Hence one of the two points p2, q2 must lie in the interior of c1, contradicting the assumption that c1 is empty. Therefore, γ2 must intersect c1 exactly twice. This implies that the 0 second portion γ2 of the curve c2 between p2 and q2 (i.e., c2 \ γ2) does not intersect γ1. Hence γ1 intersects c2 exactly once. Again, by Lemma 2.5 one of the points p1, q1 must lie in the interior of c2 (and one in the exterior of c2), a contradiction. This completes the proof of the lemma.

0 p2 γ2

γ1 q1 p1 γ2 c1

Figure 2.2: If the pseudo-discs c1, c2 are empty, the arcs γ1, γ2 cannot intersect just once.

Similar to the case of discs, we deﬁne a conﬁguration, with respect to a set P of points and a set C of pseudo-discs, to be a pair (p, c) ∈ P × C such that p lies in c.

Lemma 2.9. Let P be a set of n points in the plane. Let C be a family of m distinct pseudo-discs such that every member of C has its boundary passing through a distinct point pair p, q ∈ P . Let X denote the number of conﬁgurations in P × C. Then X ≥ m − 3n. 2.3 Pseudo-discs and Points in IR2 15

Proof. The proof proceeds by induction on m − 3n, using Lemma 2.8, and follows the same reasoning as in the proof of Lemma 2.2.

Using Lemma 2.9 and the same random sampling technique as in the proof of The- orem 2.1, we obtain the following generalization of Theorem 2.1(i):

Theorem 2.10. Let P be a set of n points in the plane. Let C be a family of m ≥ 4n distinct pseudo-discs such that every member of C has its boundary passing through a distinct point pair p, q ∈ P and such that for a given pair of points, at most one such pseudo-disc exists. Then there is a point p ∈ P that lies in Ω(m2/n2) pseudo-discs of C. The bound is asymptotically tight, in the strong sense as in Theorem 2.1.

The proof of the upper bound is delegated to Section 2.5. Theorem 2.1(ii) can also be generalized to the case of pseudo-discs:

Theorem 2.11. Let P be a set of n points in the plane, and let C be a family of m ≥ 4n distinct pseudo-discs such that every member of C has its boundary passing through a triple of points of P and for every triple of points of P there is at most one such pseudo-disc. Then there is a point in P that lies in Ω(m3/2/n3/2) pseudo-discs of C. This bound is asymptotically tight, as in Theorem 2.1.

The proof of the upper bound is delegated to Section 2.5. For the lower bound, we ﬁrst prove the following Lemma, which enables us to extend the result of Lemma 2.2.

Lemma 2.12. Let P be a set of n points and let C be a family of m distinct pseudo- discs, such that the boundary of every region c ∈ C passes through a distinct triple of points from P and has an empty interior (i.e., no point of P lies in the interior of c) and such that for any triple of points, at most one such pseudo-disc exists. Then m ≤ 2n − 4.

Proof. The proof is an easy consequence of the following claim: For a given pair p, q ∈ P , there are at most two regions in C that are spanned by both p and q. Indeed, assume to the contrary that there are three such regions c1, c2, c3 ∈ C. Each such region is spanned by both p and q and by another point of P . Denote those points, respectively, by r1, r2 and r3. Denote by γi the portion of the boundary curve of ci that is delimited by p and q and contains ri, for i = 1, 2, 3; See Figure 2.3 for an illustration. Since the pseudo-discs c1 and c2 intersect at points p and q, it follows that γ1 is either fully interior or fully exterior to c2 (except for the endpoints p and q). However, since c2 has an empty interior and γ1 contains r1, γ1 must be exterior to c2. Similarly, γ2 is exterior to c1. This is easily seen to imply that the union of γ1 and γ2 is a closed Jordan curve γ, whose interior is the union of the interiors of c1 and of c2 (See Figure 2.3). Similarly, this holds for the pair γ1, γ3 and for the pair γ2, γ3. By Lemma 2.6, one of the arcs γ1, γ2, γ3 lies in the interior of the union of the two other arcs. Assume without loss of generality that this arc is γ3. Then, since γ3 contains the point r3, r3 must lie in the interior of γ1 ∪ γ2. This however implies that r3 lies in the interior of at least one of the pseudo-discs c1, c2, a contradiction. 16 Point-Selection Lemmas

Construct a graph G on the vertex set P , by connecting, for each c ∈ C, each pair of points p, q ∈ P that are consecutive along the boundary of c, by the corresponding arc γpq ⊆ c that is delimited by p and q. Arguing as in the proof of Lemma 2.8, each pair of edges of G cross an even number of times, so G is planar. By what we have just shown, each edge of G has multiplicity at most two, so the number of edges of G is at most 6n − 12. On the other hand, this number is at least 3m, by construction, so we have 3m ≤ 6n − 12, or m ≤ 2n − 4, as asserted.

r1 q r2

γ1 γ2 p c1

Figure 2.3: γ1 is the portion of the closed Jordan curve ∂c1 between p and q that contains the point r1. Similarly, γ2 is the portion of ∂c2 that contains r2.

An immediate consequence of Lemma 2.12 is the following bootstrapping lemma:

Lemma 2.13. Let P be a set of n points in the plane. Let C be a family of m pseudo- discs as in Theorem 2.11. Let X denote the number of conﬁgurations in P × C. Then X ≥ m − 2n.

Proof. The proof proceeds by induction on m − 2n, using Lemma 2.12, and follows the same reasoning as in the proof of Lemma 2.2.

An application of the same sampling technique as in Theorem 2.1 completes the proof of Theorem 2.11. 2 One can use the same proof techniques developed in this section to obtain the following similar results on points “missing” many regions:

Theorem 2.14. Let P be a set of n points in the plane and let C be a family of m ≥ 4n distinct pseudo-discs. (i) If the boundary of every region in C contains a point pair in P and for any pair of points at most one such boundary exists, then there is a point p ∈ P that lies in the exterior of Ω(m2/n2) pseudo-discs of C. (ii) If the boundary of every curve in C passes through a distinct triple of points in P and for every triple of points, at most one such boundary exists, then there is a point p ∈ P that lies in the exterior of Ω(m3/2/n3/2) pseudo-discs of C. 2.3 Pseudo-discs and Points in IR2 17

2.3.1 Yet Another Approach for Planar Selection Lemmas In what follows we present a different and more powerful tool to obtain the following theorem. Theorem 2.15. Let P be a set of n points in the plane. Let C be a family of m ≥ 4n simply connected Jordan regions, such that the complexity of the union of any l regions of C is O(l). Assume that the boundary of each region of C passes through a distinct point pair of P . Then there exists a point q (not necessarily in P ) that is covered by Ω(m2/n2) regions of C. Proof. Let G be the graph whose vertices are the points in P and whose edges are the m point pairs (p, q) for which there is a region Cp,q ∈ C whose boundary passes through p and q. We embed G in the plane, so that the edge (p, q) is drawn along one of the two possible portions of ∂Cp,q delimited by p and q, which we choose arbitrarily 3 2 and denote it by γpq. By the Crossing Lemma, there are at least Ω(m /n ) crossing pairs of edges. Assume that no point in the plane is covered by more than k regions of C. We want to show that k = Ω(m2/n2). Since all vertices of the arrangement of the regions of C are covered by at most k regions, the number of vertices is bounded by O(mk). Indeed, using the Clarkson-Shor technique [CS89] (see also [Sha91]), the number of vertices of the arrangement A(C) that are covered by at most k regions of C is bounded by O(k2U(m/k)), where U(i) is the maximum number of vertices in an arrangement of i regions of C that are covered by 0 other regions, which is exactly the complexity of the union of these regions. By our assumption U(m/k) = O(m/k). Thus we have a lower bound of Ω(m3/n2) on the number of vertices of the arrangement A(C) and an upper bound of O(mk). Combining the two bounds, we have k = Ω(m2/n2), as asserted. Remark: We first observe that Theorem 2.15 implies a weaker version of Theorem 2.10. Indeed, since the union of any m pseudo-discs is known to have linear complexity (see [KLPS86]), we are guaranteed that in the setting of Theorem 2.10 there is always a point that is covered by Ω(m2/n2) pseudo-discs. Theorem 2.15 is weaker than Theorem 2.10 in the sense that the point guaranteed to be heavily covered is not necessarily from the set P . However, the technique of the proof is stronger in the sense that it can also be applied to more general Jordan regions (not necessarily pseudo-discs), and can also be extended to cases where the complexity of the union is slightly super-linear. Indeed, for example, we can give non-trivial bounds on points that are covered by convex “fat” regions [ES00], “fat” triangles [MPS+94], or (α, β)-covered objects [Efr99]. Definition 2.16. A convex object B in IRd is said to be α-fat if there exists two balls D,D0 such that D ⊂ B ⊂ D0 and the ratio between the radii of D0 and D is at most α, where α ≥ 1. Note that balls are 1-fat. Theorem 2.17. (Matouˇseket al. [MPS+94]): The maximum union complexity of n α-fat triangles in the plane is O(n log log n), where the constant of proportionality depends on α. 18 Point-Selection Lemmas

The constant of proportionality has recently been improved by Pach and Tardos 1 1 [PT00] to O( α log α ). Deﬁnition 2.18. [Efr99]. A planar object c is (α, β)-covered if the following holds: (i) c is simply connected, and (ii) for any point p ∈ ∂c we can place a triangle ∆ fully inside c, such that p is a vertex of ∆, each angle of ∆ is at least α, and the length of each edge of ∆ is at least β times the diameter of c.

It is not hard to see that any α-fat region is also (α0, β0)-covered, for appropriate constants α0, β0 that depend on α. Thus, the notion of (α, β)-covered objects is a generalization of that of α-fat objects.

Theorem 2.19. [Efr99]: Let C be a collection of n (α, β)-covered objects of constant description complexity and in general position1 in the plane. Assume that the boundaries of each pair of objects in C intersect in at most s points. Then the union complexity of 2 C is O(λs+2(n) log n log log n), where λs(n) denotes the maximum length of an (n, s)- Davenport-Schinzel sequence [SA95].

λs(n) Put βs(n) = n . It is well known that β1(n) = 1, β2(n) = 2 − 1/n, and β3(n) = O(α(n)), where α(n) denotes the inverse Ackermann function. In general βs(n) ≈ O(α(n)(s−2)/2) for s even. See [SA95] for further details concerning Davenport-Schinzel sequences.

Theorem 2.20. Let P be a set of n points in the plane and let T be a set of m ≥ 4n α-fat triangles. Assume that the boundary of each triangle passes through a distinct point pair of P . Then there exists a point in the plane (not necessarily of P ) that lies in Ω( m2 ) triangles of T , where the constant of proportionality depends on α. 2 n2 n log log m Proof. The proof is similar to the proof of Theorem 2.15, except that, in this case, the maximum number of vertices, in an arrangement of the boundaries of m α-fat triangles, that are covered by at most k other triangles, is O(km log log (m/k)). This fact is implied by Theorem 2.17 and the Clarkson-Shor technique [CS89]. Combining this bound with the lower bound Ω(m3/n2) implied by the Crossing Lemma, completes the proof of the theorem.

Theorem 2.21. Let P be a set of n points in the plane and let C be a set of m ≥ 4n (α, β)-covered objects with constant description complexity in the plane. Assume that the boundary of each object of C passes through a distinct point pair of P and that the boundaries of every pair of objects of C intersect in at most s points. Then there exists a point in the plane (not necessarily of P ) that lies in Ω( m2 ) objects 2 n2 2 n2 n2 n βs+2( m ) log ( m ) log log( m ) of C.

Proof. Similar to the proof of Theorem 2.15.

1A collection C of Jordan regions is in general position if at most two of their boundaries pass through a given point. 2.4 Balls and Points in Higher Dimensions 19

2.4 Balls and Points in Higher Dimensions

In what follows we say that for a given ﬁnite set of points P in IRd and for a collection of balls D, a ball b ∈ D is is spanned by a pair of points p, q ∈ P if the boundary of b passes through p and q and no other ball in D has its boundary passing through both p and q. Note that in general (in dimension d ≥ 2) for a given pair of points p, q there are inﬁnitely many balls whose boundaries pass through p and q.

Theorem 2.22. Let P be a set of n points in IRd and let D be a collection of m distinct balls spanned by distinct pairs of points of P . Then there exists a point (not necessarily in P ) that is covered by Ω(m2/n2) balls of D.

Deﬁnition 2.23. Let p and q be two points in IRd. The diametrical ball of the pair {p, q}, denoted δpq, is the smallest ball that contains p and q. Thus, δpq is centered at |pq| z = (p + q)/2, the midpoint between p and q, and has radius ρ = 2 , half the distance between p and q.

Lemma 2.24. Let P be a set of n points in Rd, and let C be a set of m balls, each spanned by a distinct pair of points of P . If m > cdn, for an appropriate positive constant cd that depends on d, then one of the following two cases must occur:

(1) There exists σ ∈ C that contains a point p ∈ P in its interior.

0 0 (2) There exist four distinct points p, q, p , q ∈ P such that σp,q0 , σp0q ∈ C, and the 0 diametrical ball δp0q spanned by p and q intersects the ball σpq0 in a set whose measure is at least βd times the measure of δp0q, for some absolute positive constant βd that depends on d.

Proof: We show that if no conﬁguration of type (1) arises, then one of type (2) must exist. An illustration of a conﬁguration of type (2) is shown in Figure 2.4(b). Let ∆ be a set of O(1) directions, represented as points on the unit sphere Sd−1, with the property that for any direction u there exists a direction u0 ∈ ∆ such that the angle between u and u0 is smaller than α = 1/2000 radians. Clearly, there exists such a d−1 set ∆ whose size, denoted by kd, is O(1/α ) = O(1). Put cd = 2kd, and assume that m > cdn. Let G be the graph whose vertices are the points of P and whose edges connect those pairs p, q ∈ P for which σpq ∈ C. We make G into a directed graph, by replacing each edge of G by two oppositely-oriented directed edges. For each u ∈ ∆, let Gu denote the subgraph of G consisting of all directed edges (p, q) such that the direction ~pq forms an angle at most α with u. {Gu}u∈∆ is a covering of G by kd (not necessarily edge-disjoint) directed graphs. Since G has more than 4kdn edges, there exists u ∈ ∆ such that Gu has more than 4n edges. Color at random each point of P red or blue (with equal probabilities), and ∗ consider the bipartite subgraph Gu of Gu consisting of all directed edges that emanate ∗ from a blue point to a red point. The expected number of edges of Gu is more than ∗ 4n/4 = n, so there exists a coloring for which the resulting Gu has at least n + 1 edges. 20 Point-Selection Lemmas

∗ For each blue or red vertex p of Gu, erase from the graph the edge (p, q) (or (q, p)) incident to p for which the Euclidean length |pq| is the largest (if the points are not in general position, erase only one such edge). We erase at most n edges. Let ~pq be a surviving edge, with p blue and q red. By construction, there exist another blue point 0 0 ~ 0 ~0 ∗ 0 p and another red point q , such that pq and p q are edges of Gu and |pq | ≥ |pq|, |p0q| ≥ |pq|. Suppose, without loss of generality, that |pq0| ≥ |p0q|. See Figure 2.4(a).

q0 q0

q q c

v v u p p σpq0 δp0q p0 p0

(a) (b)

0 0 ∗ Figure 2.4: The three edges (p, q ), (p, q), (p , q) in Gu.

Choose γ = 0.01. We distinguish between two cases: Case (i) |pq0| > (1 + γ)|pq|: The angles between any pair among the three vectors x = pq~ 0, y = ~pq, z = p~0q is at most 2α. Put x = |x|, y = |y|, z = |z|. Let c denote the 0 center of σpq0 , and let R denote its radius. Consider the plane π spanned by c, p, q . By assumption, q lies outside σpq0 . ∗ Let q denote the point that lies on the shorter circular arc of π ∩ σpq0 at distance |pq| from p (q∗ exists because |pq0| > |pq|) . The angle θ = ∠q0pq∗ is smaller than or equal to the angle ∠q0pq ≤ 2α. To see this, refer to Figure 2.5, and let s be the point of intersection between q∗c and pq0. The ball centered at s and having radius |sq∗| is ∗ fully contained in σpq0 . This implies that |sq| ≥ |sq |. Comparing the two triangles spq and spq∗, we conclude that ∠spq ≥ ∠spq∗, as asserted.

∗ s q q

c θ p

Figure 2.5: Showing that ∠spq ≥ ∠spq∗.

Clearly, R is the radius of the circumcircle of the triangle q0pq∗, so we have, by the 2.4 Balls and Points in Higher Dimensions 21

Sine Theorem, |q0q∗| R = . 2 sin θ We have γ |q0q∗| ≥ |pq0| − |pq∗| = |pq0| − |pq| ≥ |pq0|. 1 + γ

w q c

v p t σpq0 δp0q p0

Figure 2.6: The interaction between σpq0 and δp0q.

Hence, γ R ≥ |pq0|. (2.1) 2(1 + γ) sin 2α

We now turn to estimate the measure of the portion of the boundary of δp0q that lies 0 inside σpq0 . Let v denote the center of δp0q; that is, the midpoint of the segment p q. The portion under consideration is a spherical cap, whose measure (as a fraction of the total measure of the boundary of δp0q) depends only on its central angle ϕ subtended at v. This angle in turn is twice the angle at v of the triangle cvw, shown in Figure 2.6, where w is any point on ∂σpq0 ∩ ∂δp0q, which thus satisﬁes |cw| = R and |vw| = r = radius of δp0q. Put |cv| = R + t. We may assume t ≥ 0, for otherwise the angle ϕ only gets larger; see Figure 2.6. By the Cosine Theorem, we have

(R + t)2 + r2 − R2 2Rt + t2 + r2 t r2 − t2 t r cos ϕ = = = + ≤ + . (2.2) 2r(R + t) 2r(R + t) r 2r(R + t) r 2R

The fraction t/r is estimated as follows.

|vc|2 = | ~vp + ~pc|2 = |vp|2 + |pc|2 + 2 ~vp · ~pc.

√ x Hence (using the inequality 1 + x ≤ 1 + 2 ), µ ¶ µ ¶ |vp|2 2 ~vp · ~pc 1/2 |vp|2 ~vp · ~pc |vc| = R + t = R 1 + + < R 1 + + , R2 R2 2R2 R2 so |vp|2 ~vp · ~pc t < + , 2R R 22 Point-Selection Lemmas

and t |vp|2 ~vp · ~pc < + . (2.3) r 2rR rR Since ∠vqp ≤ 2α, the side vp cannot be the longest in the triangle pvq. Moreover, |vq| = r, by deﬁnition, and |pq| ≤ |p0q| = 2r. Hence |vp| ≤ max {|pq|, |vq|} ≤ 2r. 1 We have ~vp = 2 z − y. Hence ¯ ¯ ¯ ¯ ¯1 ¯ ¯R|p0q| ¯ | ~vp · ~pc| = ¯ z · ~pc − y · ~pc¯ = ¯ cos ∠(p~0q, ~pc) − R|pq| cos ∠( ~pq, ~pc)¯ ≤ ¯2 ¯ ¯ 2 ¯ ³ ´ rR cos ∠(p~0q, ~pc) + 2 cos ∠( ~pq, ~pc) . Substituting everything in (2.3), we obtain t 2r ³ ´ < + cos ∠(p~0q, ~pc) + 2 cos ∠( ~pq, ~pc) . r R

0 ~ 0 π Denote the central angle ∠pcq by 2ψ. The angle between ~pc and pq is thus 2 − ψ. Since the angles between ~pq and pq~ 0 and between p~0q and pq~ 0 are both at most 2α, it follows that π ∠(p~0q, ~pc), ∠( ~pq, ~pc) ≥ − ψ − 2α, 2 and thus cos ∠(p~0q, ~pc), cos ∠( ~pq, ~pc) ≤ sin(ψ + 2α). We have sin ψ = |pq0|/(2R), so |pq0| sin(ψ + 2α) ≤ sin ψ + sin 2α ≤ + sin 2α. 2R We thus have µ ¶ t r 5r |pq0| cos ϕ ≤ + < + 3 + sin 2α . r 2R 2R 2R Since r = |p0q|/2 ≤ |pq0|/2, we obtain 11|pq0| 11(1 + γ) sin 2α 11 + 17γ cos ϕ ≤ + 3 sin 2α ≤ + 3 sin 2α < sin 2α < 3/4, 4R 2γ 2γ say, by our choice of α and γ. We have thus shown that the central angle of the cap of δp0q inside σpq0 is at least 2 arccos 3/4, so p, q0, p0, q form a conﬁguration of type (2), with an appropriate constant βd . Case (ii): |pq0| ≤ (1 + γ)|pq|: In this case we have

|pq| ≤ |p0q| ≤ |pq0| ≤ (1 + γ)|pq|. (2.4)

This says, informally, that the three vectors ~pq, pq~ 0, p~0q are nearly the same. One difference between the two cases is that now we can no longer claim that the radius R of σpq0 must be large (cf. Figure 2.6). Instead, we tackle the problem in a diﬀerent way. 2.4 Balls and Points in Higher Dimensions 23

q0 q y x z v θ

p p0

Figure 2.7: Case (ii) of the proof.

Let v denote, as above, the midpoint of p0q, and refer to Figure 2.7. We show that the angle θ = ∠vpq0 is small (informally, this is because v is close to the midpoint of pq0). We have |vp||pq0| cos θ = ~pv · pq~ 0. 1 Since ~pv = y − 2 z, we have 1 1 1 1 |y − z|x cos θ = (y − z) · x = xy cos θ − xz cos θ ≥ xy cos 2α − xz, 2 2 1 2 2 2

where θ1 is the angle between x and y and θ2 is the angle between x and z, both at most 2α. We also have 1 z2 ³ z ´2 ³ z ´2 |y − z|2 = y2 + − y · z = y − + yz(1 − cos θ ) ≤ y − + 2yz sin2 α. 2 4 2 3 2

where θ3 ≤ 2α is the angle between y and z. Recall that y ≤ z ≤ x ≤ (1 + γ)y. Hence we have µ ¶ 1 1 1/2 z |vp| = |y − z| ≤ z + 2 sin2 α ≤ (1 + 4 sin2 α) = r(1 + 4 sin2 α). 2 4 2 We thus get, by our choice of α and γ,

z z cos 2α − z y cos 2α − 2 1+γ 2 2 cos 2α − 1 − γ cos θ ≥ 1 ≥ z 2 = 2 ≥ 0.98. |y − 2 z| 2 (1 + 4 sin α) (1 + γ)(1 + 4 sin α) Returning to the notation R, r, t, ϕ of case (i), we note that t (which, as above, can be assumed to be nonnegative) is the distance from v to σpq0 , and is thus smaller than the distance from v to pq0. The preceding calculations easily imply that this distance is attained at an interior point of pq0 (somewhere near its midpoint), so the distance is |vp| sin θ ≤ r sin θ(1 + 4 sin2 α). (To be precise, it suﬃces to verify that r sin θ(1 + 4 sin2 α) ≤ x, which follows easily by our choice of α and γ.) Using (2.2), we thus get t r 1 cos ϕ < + ≤ sin θ(1 + 4 sin2 α) + , r 2R 2 where the latter inequality follows by noting that 2R ≥ x, and r/x = z/(2x) ≤ 1/2. Hence, by our choice of α and γ, we have cos ϕ ≤ 3/4, say. 24 Point-Selection Lemmas

We have thus shown that in this case too the central angle of the cap of δp0q inside σpq0 is at least 2 arccos 3/4, so p, q0, p0, q form a configuration of type (2) with the appropriate βd. This completes the proof of the lemma. 2 Let X1 denote the number of configurations of type 1, i.e., pairs (p, σ) ∈ P × C where p lies in the interior of σ, and let X2 denote the number of configurations of type 2, i.e., pairs (σ1, σ2) ∈ C × C, spanned by four distinct points of P , where σ1 cuts off the diametrical ball δ2 corresponding to σ2 a cap whose measure is at least βd times that of δ2. Lemma 2.24 implies that X1 + X2 ≥ m − cdn. This is proven by induction on m − cdn, similar to the arguments in the preceding proofs. Specifically, the claim holds trivially for m − cdn ≤ 0. Suppose it holds for m − cdn ≤ k − 1 and consider the case m − cdn = k > 0. Lemma 2.24 implies that X1 + X2 > 0. If X1 > 0, we take a type 1 configuration (p, σ), and remove σ from C, reducing m by 1 and X1 + X2 by at least 1, so the claim follows by induction, as above. If X2 > 0, we take a type 2 configuration (σ1, σ2), remove σ2 from C, and conclude by induction, as above. Assume now that m ≥ 2cdn; otherwise, the lower bound of the theorem follows trivially. The random sampling argument used in the preceding subsections leads to 3 4 2 the inequality X1p + X2p ≥ mp − cdnp, or 2 3 X1p + X2p ≥ mp − cdn,

for any 0 < p ≤ 1. Choose p = 2cdn/m (by assumption, p ≤ 1), to obtain 4c2n2 8c3n3 d X + d X ≥ c n. m2 1 m3 2 d

Hence, one of the terms in the left-hand side is at least cdn/2, implying that m2 m3 either X1 ≥ or X2 ≥ 2 2 . 8cdn 16cdn In the former case, the pigeonholeµ principle¶ implies that there exists p ∈ P that lies in m2 m2 the interiors of at least 2 = Ω 2 balls of C. In the latter case, the pigeonhole 8cdn n principle implies that there exists σpq ∈ C whose correspondingµ diametrical¶ ball δpq m2 m2 forms conﬁgurations of type (2) with at least M = 2 2 = Ω 2 other balls of C. 16cdn n Consider the caps that these balls cut oﬀ δpq. Since the measure of each of them is at least βd times the measure of δpq, it follows that there exists a point on δpq that lies in 2 2 2 2 at least βdM = Ω(m /n ) of these caps, and thus inside Ω(m /n ) balls of C. In both cases, the bound asserted in the theorem is established. As above, the proof that the bound is tight in the worst case is delegated to Section 2.5. 2

2.4.1 Lines Stabbing Discs in IR3 Theorem 2.25. Let P be a set of n points in IR3 and let D be a set of m ≥ cn distinct (two-dimensional) discs such that every disc in D contains a distinct pair of points of 2.5 Upper Bounds 25

P on its boundary, where c is some appropriate positive constant. Then there exists a line that stabs Ω(m2/n2) discs of D.

Proof. Let {d1, . . . , dm} be the discs in D. Consider the set S = {s1, . . . , sm} of m balls, where si is the ball whose center is the center of di and whose radius is the radius of di, for i = 1, . . . , m (namely, si the smallest ball that encloses di). By Theorem 2.22, 3 m2 0 there is a point w ∈ IR that lies inside Ω( n2 ) balls of S. Denote by S the subset of balls of S containing w, and denote by D0 the corresponding subset of discs of D. Next, we choose a random line l passing through w by picking the orientation of the line randomly and uniformly from the unit sphere of directions. It is easy to see that 0 the probability that the line l stabs a disc di ∈ D is at least some absolute constant β > 0. Indeed, consider the (not necessarily circular) cone with apex at w formed by the union of all lines passing through w and through a point on the boundary of di. Let ci denote the center of di. Since w lies inside si, this cone has the property that any plane through the line ciw cuts the cone in a wedge with angle ≥ π/2. Hence the set 2 of directions on S that cause di to be stabbed is a convex cap κ with an interior point o with the property that every great circle through o cuts κ in an arc whose length is at least π/2. This is easily seen to imply the claim. This implies that the expected number of discs in D0 stabbed by l is at least β times the size of D0. Hence there must exist a line (through w) that stabs these many discs of D0. This completes the proof of the theorem. As above, the proof that the bound is tight in the worst case is delegated to Section 2.5.

2.5 Upper Bounds

As already asserted, the bounds in Theorem 2.1, Theorem 2.10, Theorem 2.11 and Theorem 2.22 are asymptotically tight in the following strong sense:

Theorem 2.26. (i) For any two positive integers m and n, with m > n, and for any dimension d ≥ 2, there is a set P of n points in IRd and a set D of m distinct balls such that every ball in D is a diametrical ball of some pair of points in P , and such that any point (not necessarily from P ) is covered by at most O(m2/n2) balls in D. (ii) For any m > n, there is a set P of n points in IR2 and a set D of m distinct discs, such that every disc in D passes through a distinct triple of points in P , and such that any point (not necessarily from P ) is covered by at most O(m3/2/n3/2) discs in D.

Proof. (i) Let s be some integer between 1 and n which will be determined later. Construct a collection of n/s clusters, each containing s points (we assume for simplicity, that s divides n). Place the clusters far apart in such a way that no diametrical ball, deﬁned by a pair of points from the same cluster, intersects any diametrical ball deﬁned ¡s¢ by a pair of points from any other cluster. For each cluster we take all 2 possible diametrical balls generated by pairs of points in the cluster. We want the number of ¡s¢ balls, which is 2 · n/s = (s − 1)n/2, to be equal to m. So we chose s = 2m/n + 1. ¡s¢ d Since a point can belong to at most 2 balls, we have that every point in IR is covered by at most O(m2/n2) balls. 26 Point-Selection Lemmas

For (ii), we have a similar construction, except that in each cluster we take all possible discs through triples of points from the same cluster, and that we place the clusters far apart to ensure that no two discs, constructed within two diﬀerent√ clusters, ¡ ¢ 3+ 1+24m/n intersect each other. We have a total of s · n/s discs. Choosing s = = 3 2 ¡ ¢ 1/2 1/2 s Θ(m /n ), the number of discs is m. Since no point is covered by more than 3 = O(s3) discs, we obtain the desired upper bound. (As stated, the constructions do not apply to all values of m and n. However, by slightly modifying the choice of S and the construction itself, we can extend the bound for all values of m and n.) Remark: Tightness of Theorem 2.25 can be shown by a similar construction involving n points in R3 and m diametrical discs, each of which passes through a pair of the given points within the same cluster. The n/s clusters should be arranged such that no line stabs more than two clusters (namely, the set of discs stabbed by any given line is generated within at most two clusters). This is easily done by taking n/s points in convex position (say, on a unit sphere) and replacing each such point p with a cluster of s points all of which are “very close” to p. Hence, no line stabs more than two clusters, ¡s¢ and therefore at most 2 · 2 discs. Choosing, as above, s = 2m/n + 1, we have that no line can stab more than O(m2/n2) discs.

2.6 Axis-Parallel Rectangles

Let P be a set of n points in the plane. For simplicity we assume that no pair of points have the same x-coordinate or the same y-coordinate. Let R be a set of m axis-parallel rectangles, each having two points of P as opposite vertices. Lemma 2.27. If m > 4n log n then either (a) there exists a rectangle in R that contains a point of P in its interior, or (b) there exist two rectangles R1,R2 ∈ R, spanned by four distinct points of P , such that a vertex of one of them lies in the interior of the other. Proof: We assume that case (a) does not arise, and argue that case (b) must then occur. Either at least half of the rectangles in R are such that their bottom-left and top-right vertices are in P , or at least half of them are such that their bottom-right and top-left vertices are in P . Without loss of generality, assume that the former case arises, and remove from R all other rectangles. We now have |R| > 2n log n. For each + − point a ∈ P , let Ra (resp., Ra ) denote the set of all rectangles in R having a as their bottom-left (resp., top-right) vertex. Since we have assumed that case (a) does not + occur, no rectangle in Ra fully contains another such rectangle, so these rectangles can be ordered in increasing order of the x-coordinate, which is the same as the decreasing order of the y-coordinate, of their top-right corners (all of which are points of P ). In a − fully symmetric manner, the rectangles in Ra can be ordered in the same two coinciding orders. + Call a rectangle R ∈ Ra left-separated if either R is the ﬁrst in the ordered sequence + Ra , or the preceding rectangle in that sequence is such that its width (x-span) is at least − twice as small as the width of R. Similarly, we call a rectangle R ∈ Ra right-separated 2.6 Axis-Parallel Rectangles 27

− if either R is the last in the sequence Ra , or the next rectangle in that sequence is such that its width is at least twice as small as the width of R. Clearly, the number of + − rectangles that can be right-separated or left-separated in the respective sets Ra , Ra is at most 2 log n, so the number of such rectangles, over all points a ∈ P , is at most 2n log n. Since |R| is larger than this bound, R contains at least one rectangle that is neither left-separated nor right-separated. Let a, b ∈ P denote, respectively, the bottom-left and top-right vertices of R. Let R0 (resp., R00) denote the rectangle preceding (resp., + − 0 00 succeeding) R in the sequence Ra (resp., Rb ). Clearly, R and R are spanned by four distinct points of P , and the top-left vertex of R00 lies in the interior of R0 (and the bottom-right vertex of R0 lies in the interior of R00); see Figure 2.8 2

00 R

0 + 00 Figure 2.8: A non-separated rectangle R and the two adjacent rectangles R ∈ Ra ,R ∈ − Rb that realize case (b) of the lemma.

Let X1 denote the number of configurations of the form (R, a), where R ∈ R and a ∈ P are such that a lies in the interior of R. Let X2 denote the number of configurations of the form (R,R0), where R,R0 ∈ R are two rectangles that are spanned by four distinct points of P and are such that a vertex of R0 lies in the interior of R. We refer to configurations of the former (resp., latter) type as type I (resp., type II) configurations. Lemma 2.27 implies the following inequality

X1 + X2 ≥ m − 4n log n. (2.5) We apply (2.5) to a random subset of the given points and rectangles, where each point in P is chosen independently with probability p, and a rectangle is chosen when its 0 0 0 0 two spanning P -points are chosen. Let n , m ,X1,X2 denote, respectively, the expected number of points, rectangles, type I conﬁgurations, and type II conﬁgurations in the sample. We have 0 0 0 0 X1 + X2 ≥ m − 4n log n. As is easily checked, we have

0 0 2 0 3 0 4 n = np, m = mp ,X1 = X1p ,X2 = X2p . Hence, 3 4 2 X1p + X2p ≥ mp − 4np log n. 28 Point-Selection Lemmas

8n log n We assume that m ≥ 8n log n, and choose p = . Suppose first that X ≥ X p. m 1 2 Then we have 3 2 2X1p ≥ mp − 4np log n = 4np log n, or 2n log n m2 X ≥ = . 1 p2 32n log n By the pigeonhole principle, there exists a point a ∈ P that participates in at least m2 type I configurations, that is, a lies in at least that many rectangles of R. 32n2 log n Suppose next that X1 < X2p. Then we have 4 2 2X2p ≥ mp − 4np log n = 4np log n, or 2n log n m3 X2 ≥ = . p3 256n2 log2 n Again, by the pigeonhole principle, there exists a rectangle R ∈ R that participates as m2 the second component of at least type II configurations. This implies that 256n2 log2 n m2 one specific vertex of R lies in the interior of at least rectangles of R. 512n2 log2 n We have thus shown: Theorem 2.28. Let P be a set of n points in the plane, so that no pair of points have the same x-coordinate or the same y-coordinate. Let R be a set of m ≥ 8n log n axis- parallel rectangles, each having two points of P as two opposite vertices. Then there m2 exists a point v ∈ R2 that is contained in the interior of at least rectangles 512n2 log2 n of R. Theorem 2.28 also holds when the points in P may have common x-or y-coordinates, except that in this case the stabbing point may lie on the boundary of some of the stabbed rectangles. We note that a different proof of Theorem 2.28, based on certain one-dimensional selection lemmas, is given in [CEG+94]. Our proof can also be easily extended to axis- parallel boxes in any dimension, yielding an alternative proof of a similar extension obtained in [CEG+94]. We omit here the easy details of this extension. These extensions imply that there always exists a point of IRd that lies in the interiors of Ω( m2 ) of n2 log2d−2 n the given m boxes.

2.6.1 An Upper Bound We next show that the polylogarithmic factor appearing in the lower bound of Theo- rem 2.28 cannot be totally eliminated to yield the bound Ω(m2/n2). Speciﬁcally, we show:2 2We are indebted to Rom Pinchasi for an idea that led to this construction. 2.6 Axis-Parallel Rectangles 29

¡n¢ Theorem 2.29. For arbitrarily large n and m, satisfying cn log n ≤ m ≤ 2 , for an appropriate constant c, there exist sets P of n points and R ofÃm rectangles! spanned by m2 the points of P , so that no point in R2 lies in more than O rectangles of 2 n2 n log m R.

Proof: We construct sets P and R whose respective sizes n and m are deﬁned in terms of two integer parameters k and j. Let k be a ﬁxed integer. We construct P and R recursively, starting with an arbitrary set P0 of n0 = k points in general position, and with the set R0 of all axis-parallel rectangles spanned by pairs of points of P0. We have ¡k¢ m0 = |R0| = 2 . Suppose that we have already constructed Pj and Rj, for some j ≥ 0. We construct Pj+1 and Rj+1 as follows.

(1) (2) (1) (2) (i) Take two distinct copies Pj ,Pj of Pj, keep Pj intact, and shift Pj hori- zontally so that the x-spans of the two copies are pairwise disjoint. Create two (1) (2) corresponding copies Rj , Rj of Rj.

(2) (2) (ii) Next, shift the copy Pj (and, accordingly, also each rectangle in the copy Rj ) slightly upwards in the vertical direction, so that if point a lies below point b in Pj then both copies of a lie below both copies of b.

(iii) For each pair of points a, b ∈ Pj, such that a lies below b, and there are at most k − 2 points of Pj in the (open) horizontal strip spanned by a and b, create a rectangle whose opposite vertices are the ﬁrst copy of a and the second copy of (1) b. (Thus, each point of Pj , except for the k − 1 top ones, participates in k such rectangles.)

(1) (2) (iv) Take Pj+1 to be the union of Pj and Pj , and take Rj+1 to be the union of (1) (2) Rj and Rj , together with all the additional rectangles created at the preceding step.

See Figure 2.9 for an illustration of this construction.

c0 c b0

a0 a

Figure 2.9: The recursive step of the construction (shown with k = 2). 30 Point-Selection Lemmas

Put nj = |Pj| and mj = |Rj|. We have µ ¶ k n = 2n , m = 2m + kn − . j+1 j j+1 j j 2

¡k¢ (The term 2 accounts for the fewer numbers of rectangles spanned by the very top (1) points of Pj .) We thus have, as is easily veriﬁed by induction on j, µ ¶ (j + 1)kn k n = k · 2j, m = j − . j j 2 2

Let ξj denote the maximum number of rectangles in Rj that have nonempty intersection. We have µ ¶ k ξ ≤ , 0 2 µ ¶ k + 1 ξ ≤ ξ + . (2.6) j+1 j 2 (1) (2) Indeed, let v be any point in the plane. The x-spans of Pj and of Pj are disjoint, (1) and the x-coordinate of v can belong to at most one of them, say to that of Pj . Then (1) v can be contained only in rectangles belonging to the corresponding set Rj , and in rectangles created at step (iii). The number of rectangles of the latter kind is at most ¡k+1¢ (1) 2 : v can only lie in rectangles spanned by the i-th point of Pj below v (in their (2) y-order) and the `-th point of Pj above v, where i+` ≤ k +1, and the number of such ¡k+1¢ pairs is at most 2 . This establishes the recurrence (2.6), whose solution is easily seen to be µ ¶ ¡k¢ k + 1 mj + 2 ξj ≤ (j + 1) = (k + 1) · . 2 nj

For any choice of k and j, we obtain an instance of the problem with n = nj points and m = mj rectangles. It is easily seen that, by varying k and j, we can have mj vary 2 between Θ(nj log nj) (choose k = 1 for this extreme case) and Θ(nj ) (choose j = 1). An easy calculation shows that 2m 2j+1 n2 k ≈ , and ≈ , jn j m

which implies that Ã ! m k = Θ . n2 n log m Hence, the maximum number of rectangles with a nonempty intersection is at most µ ¶ Ã ! km m2 O = O , n 2 n2 n log m as asserted. 2 2.7 Open Problems 31

2.7 Open Problems

• Section 2.4 deals with point selection bounds for balls spanned by pairs of points of a finite set of points in IRd. It would be interesting to generalize the technique used there, to obtain non-trivial bounds for balls spanned by j-tuples of points (where j is a fixed integer between 3 and d). In addition, it would be nice to find a simpler proof of Lemma 2.24.

• It would be interesting to tighten the logarithmic gap between the lower and upper bounds described in Section 2.6 for axis-parallel rectangles. 32 Point-Selection Lemmas Chapter 3

Triangles and Points in IR3

3.1 Introduction

In this chapter we study problems involving convex polygons in the plane or convex polytopes in IR3 with vertices from a given point set. Let S be a set of n points in IR3 in general position (i.e., no four points in S are co-planar and no three points in S are co-linear) and let P be a set of k convex polytopes spanned by points of S (i.e., their vertices belong to S), such that the polytopes have pairwise distinct facets, that is, a triangle ∆ spanned by a triple of points a, b, c ∈ S can be a facet of at most one polytope in P. We wish to determine the maximum possible total complexity of the polytopes in P. In Section 3.4 we derive an upper bound of O(n2k1/3) on this number. This reproduces an earlier result of Aronov and Dey [AD01], using a much simpler argument, which is also based on the probabilistic proof technique used in the previous chapter. Suppose that, in addition to the original assumptions, we assume that the polytopes spanned by S have the property that for any pair of points p, q in S and any plane h that passes through p and q there is at most one polytope P ∈ P, such that P contains the edge pq on its boundary and h is tangent to P . In Section 3.4 we derive an improved upper bound of O(n3/2k1/2) on the overall complexity of such collections of k polytopes. As a warm-up exercise, we consider the planar version of this problem, and show that the maximum number of edges bounding k distinct and edge-disjoint convex polygons that are spanned by a set S of n points in the plane (i.e., each such convex polygon has its vertices from S), is Θ(nk1/2).

3.2 Polygons in Point Sets and Concave Chains in Arrangements of Lines

Let H be a set of n hyperplanes in IRd and let P be a set of k convex polytopes in the arrangement A(H). Each such polytope is a cell in an arrangement of some subset of hyperplanes of H. Assume further that the polytopes in P do not share vertices. Aronov and Dey [AD01] denote the maximum overall complexity of such a collection 34 Triangles and Points in IR3

of polytopes by Kvert(k, n, d); this is the number of faces of all dimensions that bound the polytopes in P. In the plane, Halperin and Sharir [HS92] have shown that

1/2 Kvert(k, n, 2) = Θ(nk ) , a result motivated by their analysis of certain motion planning problems with three degrees of freedom. When the polygons (or, more generally, convex polygonal chains) are further constrained not to overlap along edges (i.e., when the given polygon boundaries or polygonal chains can only cross each other transversally), the respective bound is:

Θ(nk1/3) for k ≤ n and

Θ(n2/3k2/3) for n ≤ k ≤ n2. In this section we give alternative and, as we believe, simpler, proofs for the upper bounds in both cases and for the lower bound of the ﬁrst case (which is diﬀerent and somewhat simpler than the previous construction in [HS92]). For the upper bounds we use the Crossing Lemma mentioned in Chapter 1. (Unlike the previous chapter, where we used only the proof technique of that lemma, here we use the lemma itself.)

Upper Bounds We analyze separately each of the two cases mentioned above: Case (1): The polygons are not allowed to share vertices. Case (2): The polygons are not allowed to have (partially) overlapping edges. We apply a standard dual transformation, in which points are mapped into lines and lines are mapped into points. Speciﬁcally, a point p with coordinates (a, b) is mapped to a line p∗ deﬁned by the equation y = −ax + b and a non-vertical line l with equation y = cx + d is mapped to the point l∗ = (c, d). It is easily seen that a point p lies above (resp., on, below) a non-vertical line l if and only if the line p∗ passes above (resp., through, below) the point l∗.1 In the dual setting we are faced with the following problems: We have a set S of n points and a set C of k convex polygons, or polygonal chains whose vertices belong to S. In Case (1) the constraint that the original polygons are not allowed to share vertices, transforms into the constraint that the polygons in C are not allowed to share edges. In Case (2) the additional constraint that the original set of polygons are not allowed to overlap in edges transforms to the following property: For each point p in S and for each line l through p, there is at most one polygon P in C such that l is tangent to P at p. (This follows by noting that the set of lines tangent at p to a dual convex polygon

1See the book [dBvKOS00] for a detailed description of other similar dual transformations. 3.2 Polygons in Point Sets and Concave Chains in Arrangements of Lines35

P ∗ is dual to the set of points on an edge of the original polygon P that lies on the line dual to p.) In both cases we aim to bound the overall complexity of C, which is the total number of edges of all the polygons. Denote the set of those edges by E. We apply the Crossing Lemma, mentioned in Chapter 1, to the set E of all the edges of the polygons in C. The number X of crossings between edges of E is thus at least |E|3 64n2 , assuming that |E| ≥ 4n. On the other hand, since each edge in E can cross any convex polygon boundary at most twice, the number of crossings in E is bounded by k|E|. Combining the two bounds yields the upper bound 8nk1/2 on |E|. This subsumes the alternative case |E| < 4n, and establishes the upper bound for Case (1). In Case (2) we have the following property of the sets S and C.

Lemma 3.1. The boundary of any convex polygon (or a convex polygonal chain) P can cross at most 2(k + n) edges of E.

Proof. Let P be a convex polygon. Construct an expanded homothetic copy P 0 of P , with a suﬃciently large scaling factor, so that ∂P 0 crosses no edges of E. Start shrinking P 0 until it coincides with P . We note that the number of crossings between edges of E and edges of P 0 changes only when P 0 passes through a point p in S or when a vertex of P 0 passes through an edge of E. In the latter case, the number of crossings between P and the edges of E decreases. Hence, this number can increase only when the boundary of P 0 is tangent to some polygon Q in C, at some point p ∈ S. By assumption, there exists at most one such polygon Q (unless p is the end of some polygonal chain; the total number of such events is at most 2k), and the number of edges that the boundary of P 0 crosses increases by two (or by one when p is the end of a chain). Hence, the boundary of P can cross at most 2(k + n) edges of E.

Lemma 3.1 implies that the number X of pairs of crossing edges is at most the number of polygons multiplied by (k + n); namely, X ≤ 2kn for k < n and X ≤ 2k2 for k ≥ n. Using the Crossing Lemma as above, we obtain, for k ≥ n, that either |E| ≤ 4n 3 |E| 2 2/3 2/3 2/3 2/3 or 64n2 ≤ 2k , or |E| = O(n k + n) = O(n k ). For k < n, we have that either 3 |E| 1/3 |E| ≤ 4n or 64n2 ≤ 2nk, or |E| = O(nk ). This completes the proof of the upper bound for Case (2).

Lower Bound We give the following construction for the lower bound of Case (1), which, as noted, is diﬀerent and somewhat simpler than the one given in [HS92]. For given positive integers ¡n¢ n and k ≤ 2 , we construct a set S = {p1, . . . , pn} of n points in convex position and a set C of k convex polygons with vertices from S such that all polygons in C have 1/2 distinct edges and√ such that the total complexity of the polygons in C is Ω(nk ). We construct t = k collections of polygons, in t stages, as follows: In the ﬁrst stage we construct one polygon which is the convex hull of S. In the second stage we construct two polygons, each being the convex hull of bn/2c points of S, so that one is the hull of {p1, p3,...} and the other is the hull of {p2, p4,...}. In the i-th stage we construct the 36 Triangles and Points in IR3

i convex hulls of the sets {pl, pl+i, pl+2i, . . . , pl+(bn/ic−1)i} for l = 1, . . . , i. We stop when we have constructed a total of k polygons. This means that we have at least Ω(k1/2) stages. Note that the total number of edges of the polygons in each stage is Ω(n). This implies that altogether we obtain a set of k edge-disjoint convex polygons with a total of Ω(nk1/2) edges. See Figure 3.1 for an illustration.

pn p3

p2 p1

Figure 3.1: A construction of a set S of n points and of k polygons spanned by S with a total of Ω(nk1/2) edges.

3.3 Triangles and Points in IR3

In this section we give a simple proof for the following three-dimensional analog of the Crossing Lemma.

3 Deﬁnition 3.2. Let ∆1 be a triangle in IR with vertices a, b, c and let ∆2 be a triangle with vertices a, d, e (i.e., ∆1 and ∆2 share a vertex). We assume that the points a, b, c, d, e are in general position (i.e., no four of them are co-planar). We say that two such triangles cross if their relative interiors cross. See Figure 3.2. Note that if ∆1 and ∆2 cross then either the straight line segment de crosses the relative interior of ∆1 or the straight line segment bc crosses the relative interior of ∆2. Theorem 3.3. Let S be a set of n points in IR3 and let T be a set of t ≥ 2n2 distinct triangles whose vertices belong to S. Let X denote the number of crossing pairs of triangles of T (in the above sense). Then X = Ω(t3/n4).

Proof. For a point p ∈ S let tp denote the number of triangles of T that are incident to p (i.e., that have p as one of their vertices). Let X denote the number of pairs of 4 2 triangles that cross. First we claim that X ≥ t − 3 n . This is shown by induction on 4 2 4 2 t − 3 n and is similar to the proof of Lemma 2.2. Indeed, notice that if t − 3 n ≥ 0, P 2 then for at least one of the points p ∈ S, tp ≥ 4n, since p∈S tp = 3t ≥ 4n . We draw a small two-dimensional sphere S centered at p and project all triangles incident to p 3.3 Triangles and Points in IR3 37

onto S centrally from p. In this projection we have a drawing of a graph on a set of n − 1 points (these are the projections of all the n − 1 points of S \{p} onto S) and a set of tp great circular arcs (these are the projections of the triangles incident to p onto S) connecting pairs of points on S. Since tp ≥ 4n, this graph cannot be planar, so there must be a pair of arcs that cross. It is easily seen that such a crossing corresponds to two triangles incident to p that cross. Hence X ≥ 1 and the proof proceeds by removing a triangle involved in a crossing, and using induction on the remaining set. Next, we take a random sample S0 ⊂ S by picking each point of S independently with probability r (to be determined later). Let T 0 be the set of triangles all of whose vertices belong to S0 and let X0 be the number of pairs of triangles of T 0 that cross. Apply the above inequality to the expectations of the random variables |S0|2 , |T 0| and X0. It is easy to see that E[X0] = r5X and E[|T 0|] = r3t. We also have

h i ·µ 0 ¶¸ µ ¶ 2 |S | n E |S0| = 2 E + E[|S0|] = 2 r2 + nr ≤ n2r2 + nr. 2 2

Hence 4 4 r5X ≥ r3t − r2n2 − rn 3 3 or t 4n2 4n X ≥ − − . r2 3r3 r4 2 t3 Choosing r = 2n /t (by assumption, r ≤ 1), we obtain that X ≥ 24n4 , as is easily ¡n¢

checked (using the fact that t ≤ 3 ).

Figure 3.2: Two crossing triangles (with a common vertex).

Corollary 3.4. Let S be a set of n points in IR3 and let T be a set of t ≥ 2n2 distinct triangles whose vertices belong to S. Then there exists a line that stabs Ω(t3/n6) triangles of T .

Remark: This fact is known (see [DE94]). However, our proof is diﬀerent and simpler, and is another manifestation of the general probabilistic theme related to the proof of the Crossing Lemma and recurrent in the ﬁrst part of the thesis. 38 Triangles and Points in IR3

Proof. As a matter of fact we show that there exists a segment connecting two points of S that stabs Ω(t3/n6) triangles of T . Let k be the maximum number of triangles of T that such a segment stabs. By Theorem 3.3 we know that X, the number of crossing pairs of triangles of T is Ω(t3/n4). Fix an edge e = pq with endpoints in S. For each triangle ∆ = abc that it crosses, e can contribute at most three crossings to X, namely a crossing between abc and apq, between abc and bpq, and between abc and cpq. Since ¡n¢ ¡n¢ there are at most 2 possible such edges then we have an upper bound of 2 ·3k on X. Combining the lower and upper bounds on X we have k = Ω(t3/n6). This completes the proof of the corollary.

Remark: In Chapter 4 we prove that in the special case when all triangles of T are halving triangles, then X = Ω(t2/n) and hence k = Ω(t2/n3) (note that t2/n3 À t3/n6 when t = o(n3)). This improved bound holds in more general situations, and is also exploited in Theorem 3.7 below.

3.4 Polytopes Spanned by Point Sets

Let S be a set of n points in IR3 in general position, that is, no four points of S are coplanar. Let P be a collection of k polytopes spanned by S and having distinct faces, i.e, each polytope in P is the convex hull of a subset S0 of S and each triangle ∆ that is spanned by three points in S is a facet of at most one polytope in P. (Note that our general position assumption on S implies that the polytopes in P are all simplicial, and their faces are indeed triangles.) Our goal is to bound the overall complexity of P, that is, the total number of facets of all the polytopes in P.

Theorem 3.5. The overall complexity of a collection P of k polytopes with distinct facets that are spanned by a set S of n points in IR3 in general position, is O(n2k1/3).

Proof. Denote the set of triangles which are facets of some polytope in P by T . By Theorem 3.3, there exists a line that crosses Ω(t3/n6) triangles of T . Since a line can cross each polytope boundary at most twice, this quantity is at most 2k. Combining the two bounds completes the proof of the theorem.

Remark: Note that a trivial bound on the overall complexity of P is O(nk). This bound is smaller than the bound just derived when k = O(n3/2). Hence Theorem 3.5 is signiﬁcant only for larger values of k. As noted above, this theorem has already been established by Aronov and Dey [AD01], using a more complicated proof. Next, we turn to analyze the following more restricted class of polytopes spanned by a point set in IR3. In addition to the requirement that the polytopes in P have distinct facets, we also require that for any pair of points p, q in S and any plane h that passes through p and q there is at most one polytope P ∈ P, such that P contains the edge pq on its boundary and h is tangent to P . We refer to this property as the tangency property. 3.4 Polytopes Spanned by Point Sets 39

We show that if T is the collection of triangles that are facets of such a collection P of polytopes, then there exists a line that crosses Ω(t2/n3) triangles of T . Since, as above, such a line can cross each polytope boundary at most twice, this quantity is at most 2k. Combining these bounds yields:

Theorem 3.6. Let S be a set of n points in IR3 in general position and let P be a collection of k polytopes spanned by points of S such that the polytopes have distinct facets and for every edge pq and any plane h that passes through pq, h can be tangent to at most one polytope in P that contains the edge pq on its boundary. Then the overall complexity of P is O(n3/2k1/2).

Theorem 3.6 is an immediate consequence of the following lower bound, when combined with the preceding observations.

Theorem 3.7. In the setup of Theorem 3.6, the number X of crossing pairs of facets of the polytopes in P satisﬁes X = Ω(t2/n).

Proof. The proof is a variant of the proof of Lemma 4.8 and we will present it in Section 4.2.3 of the following chapter.

Remark: Similar to a preceding observation, the trivial bound O(nk) beats the bound of Theorem 3.6 when k ≤ n.

3.4.1 Lower Bound

We give a simple construction of a set S of n points in IR3 and a collection P of k ≥ n polytopes spanned by S with distinct facets such that the total complexity of P is Ω(n3/2k1/2). (This bound has to be compared with the upper bound of Theorem 3.5 and not with that of Theorem 3.6, since the polytopes that we construct do not have the additional tangency property assumed in Theorem 3.6.)

Let S0 be a set of n/2 points on the unit circle in the plane z = 0. We construct 1/2 a collection C0 of l = 4k/n polygons in S0 with a total complexity Ω(nl ), using the construction described in Section 3.2. The points of S0 are not in general position, so we slightly perturb each of them in the z-direction to put them in general position. + © ª Next, we place a set S1 = u1, . . . , un/4 of n/4 points near the positive z-axis, keeping − © ª them in general position, and a set S1 = v1, . . . , vn/4 of n/4 points in general position near the negative z-axis. For each polygon P in C0 and for each i = 1, . . . , n/4, we construct a polytope Pi equal to the convex hull of P ∪ {ui, vi}. Altogether we obtain + − 3 a set S = S0 ∪ S1 ∪ S1 of n points in general position in IR , and a collection P of l · n/4 = k convex polytopes spanned by S, which, as is easily veriﬁed, have distinct facets (we use here the fact that the origin lies in the interior of each of the polygons 1/2 3/2 1/2 in C0). The overall number of facets of these polytopes is Ω(nl ) · n/2 = Ω(n k ). This completes the lower bound construction. 40 Triangles and Points in IR3

3.4.2 Open Problems Both bounds of Theorem 3.5 and 3.6 are not known to be tight. We conjecture that the bound in Corollary 3.4 is not tight and that the correct lower bound there is Ω(t2/n3), which, in view of the lower bound just given, is tight in the worst case. This however does not yield an improvement of Theorem 3.6, and an open problem is to improve the bound, or to provide a matching lower bound. Chapter 4 k-Sets in IR3

4.1 Introduction

Let S be a set of n points in IRd.A k-set of S is a subset S0 ⊂ S such that S0 = S ∩H for some halfspace H and |S0| = k. The problem of determining tight asymptotic bounds on the maximum number of k-sets in an n-element set in IRd is one of the most intriguing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [CSY87, EW85], the problem has caught the attention of computational geometers as well [ACE+91, DE94, EVW97, Sha91, Wel86]. A close to optimal solution for the problem remains elusive even in the plane. The√ best asymptotic upper and lower bounds in the plane are O(nk1/3) [Dey98] and n · 2Ω( log k) [T´ot01],respectively. In this chapter we obtain the following result:

Theorem 4.1. The number of k-sets in a set of n points in IR3 is O(nk3/2).

This result improves the previous best known asymptotic upper bound of O(nk5/3)

(see Dey and Edelsbrunner [DE94] and Agarwal et al. [AACS98]). The best known√ asymptotic lower bound for the number of k-sets in three dimensions is nk · 2Ω( log k) (obtained by lifting the planar construction to three dimensions; see [T´ot01]). The results of this chapter are joint work 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. 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 this chapter.

4.2 k-Sets and Triangles in IR3

4.2.1 An Overview of Our Technique (a) The paper [AACS98] presents a general technique, based on random sampling, for transforming an upper bound on the number of k-sets that is independent of k to a bound that does depend on k. Our main thrust will thus be to establish the upper bound 42 k-Sets in IR3

O(n5/2) for the number of k-sets. This, combined with the technique of [AACS98], will imply Theorem 4.1. (b) We assume that the set S is in general position, meaning that no four points in S lie in a common plane. Applying a small perturbation to the points of any set S yields a set of points in general position and the number of k-sets does not decrease. (c) We consider the set T of halving triangles spanned by S: A triangle ∆ = abc, with vertices a, b, c ∈ S is a halving triangle if the plane containing ∆ has the same number of points of S on either side. (Note that n has to be odd for halving triangles to exist, and we will indeed assume, without loss of generality, that n is odd.) We show that |T | = O(n5/2). This implies that the number of k-triangles, for any k, is also bounded by O(n5/2), where a k-triangle is a triangle ∆ spanned by three points in S with exactly k points of S on one side of the plane containing ∆. Indeed, choose a direction d not contained in the plane of any k-triangle and add |n−3−2k| extra points to S far enough in the direction d or −d. Each k-triangle in S turns into a halving triangle in one of the two resulting configurations. It is well known [AAHP+98], that the O(n5/2) bound on the number of k-triangles for any k carries over to the same bound on the number of k-sets. (d) All the previous approaches to bounding the number of k-sets are based on (the 3-dimensional extension of) Lovász’Lemma [BFL90]: Any line crosses (the relative interiors of) at most O(n2) halving triangles. The preceding techniques aimed to derive a general lower bound for the number of such crossings. Specifically, they showed that for any collection of t triangles spanned by the points of S there exists a line that crosses many triangles, where the best lower bound for this number of crossings is Ω(t3/n6) ([DE94]; see also Corollary 3.4). Combining this lower bound with the upper bound provided by Lovász’Lemma, one obtains an upper bound of O(n8/3) for the number of k-sets. (e) In contrast, our technique focuses on the specific set T of halving triangles, and exploits the structure of this set. The main property of this set, which is also used in deriving Lovász’Lemma, is the antipodality property, which we re-establish rigorously in Lemma 4.4 below. Informally, it asserts that the halving triangles with a common edge pq alternate sides, as we rotate a plane containing pq. See Figure 4.1 for an illustration of this property. This is the only property of the set T that is needed in the proof. (f) As in Chapter 3, our technique only considers interaction between pairs of triangles of T with a common vertex. Specifically, we consider crossings between such pairs of triangles, where two triangles pab and pcd cross each other if their relative interiors have a nonempty intersection (in this case p is the only common vertex of these triangles). (g) Our proof proceeds by deriving both an upper bound and a lower bound on the number of triangle crossings (of the above special type) in T . The upper bound is O(n4) and it is an easy consequence of Lovász’Lemma in 3-space. The lower bound is Ω(t2/n), and is proven using arguments that extend those that were used in [Dey98] for the analysis of k-sets in the plane. These upper and lower bounds immediately yield the desired bound on the number of k-sets. 4.2 k-Sets and Triangles in IR3 43

2 9

7 4 5 6 pq

1 3 8

Figure 4.1: The antipodality property of halving triangles: The common edge pq is shown head-on, as a point; as we rotate a plane containing pq we encounter the other vertices of these triangles in the order shown.

4.2.2 Proof of the Theorem Let n be odd, let S be a set of n points in IR3 in general position, and let T be the set of all halving triangles of S. Put t = |T |. Let X denote the number of crossing pairs of triangles in T , where, as in Defini- tion 3.2, we say that two triangles ∆1, ∆2 ∈ T cross if ∆1 and ∆2 share exactly one vertex, say p, and the edge opposite to p in one of the triangles crosses the other triangle (this is equivalent to the definition given in Section 4.2.1). The following extension of the two-dimensional Lovász’Lemma [ELSS73] has been derived in [BFL90] and used in [ACE+91, BFL90, DE94]. We say that a line crosses a triangle if it intersects the triangle but not any of its edges. One can prove this lemma using the Antipodality Lemma below by translating a line from infinity to the given location, and by observing how the number of triangles crossed by the line changes as it moves—this number changes only when the line crosses a segment connecting two points and then it changes by ±1.

Lemma 4.2. [BFL90, Lov71] A line crosses fewer than n2/4 halving triangles.

As a consequence we obtain:

Lemma 4.3. The number X of crossing pairs of halving triangles for a set S as above is less than 3n4/8.

Proof. Fix an edge e = pq with endpoints in S. This edge crosses fewer than n2/4 triangles. For each triangle ∆ = abc that it crosses, e can contribute at most three crossings to X, namely a crossing between¡ ¢abc and apq, between abc and bpq, and n 4 between abc and cpq. Since there are only 2 edges, we have in total fewer than 3n /8 crossings. 44 k-Sets in IR3

The following well-known lemma, which is the basis for the 3-dimensional version of Lov´asz’Lemma (see, e.g., [ACE+91, BFL90]), will be crucial for our analysis. We include a proof for the sake of completeness.

Lemma 4.4 (Antipodality Lemma). Let p, q ∈ S and let Tpq denote the subset of all triangles in T incident to both p and q. Rotate a halfplane h, bounded by the line ` passing through p and q, about `; h meets the triangles in Tpq in a cyclic order. Let ∆ 0 and ∆ be two consecutive elements of Tpq in this cyclic order, let W be the wedge swept by h as it rotates from ∆ to ∆0, and let W 0 denote the antipodal wedge, emanating from ` and bounded by the same pair of planes. Then there is a unique ‘antipodal’ triangle 00 0 ∆ ∈ Tpq contained in W . Proof. Consider the halfplane h rotating about pq. If during the rotation h contains a halving triangle pqr and the next such triangle is pqr0, then as h leaves r the plane containing h has one more point of S on its side containing r than on the opposite side. Just before reaching r0 the plane containing h has one more point on its side containing r0 than on the opposite side containing r. Since the diﬀerence between the number of points of S contained in the two sides changes by one each time the plane reaches or leaves a point of S, there must be a position in between when the diﬀerence is zero. At that point the plane containing h contains a halving triangle from Tpq, but since ∆ and ∆0 are consecutive, this halving triangle is not contained in h but in the opposite halfplane and therefore in W 0. The uniqueness of this antipodal triangle is a consequence of the existence proof: If 0 there were two or more antipodal triangles in Tpq for ∆ and ∆ , then one could choose two consecutive ones and this pair of two consecutive elements of Tpq would have no antipodal triangle.

Remarks: (a) Note that |Tpq| must be odd to satisfy the assertion of the lemma, unless Tpq is empty. It is easy to show that Tpq is not empty for any edge pq. If Tpq has a single element the assertion of the lemma holds automatically. In all other cases the lemma implies that any halfspace with p and q on its boundary contains at least one element of Tpq. (b) We say that a collection T of triangles that is spanned by S is antipodal if it satisfies the property in Lemma 4.4. Inspecting the foregoing proof, it is easily verified that it also applies to any antipodal collection T . Hence any such collection can have at most O(n5/2) triangles. As a matter of fact, this also holds for weakly antipodal collections T , meaning that, for each edge pq, the antipodality property holds for all but a constant number of consecutive pairs of triangles in Tpq. We fix a coordinate frame and assume that no horizontal plane (i.e., one parallel to the xy plane) contains more than one point of S. We further assume that the plane of no triangle in T is parallel to the y-axis. This can be achieved by a suitable rotation. Fix a point p ∈ S, and let Tp denote the set of triangles in T that are incident to p. Let hp be the horizontal plane passing through p. Let πp be any horizontal plane above p. Clip each triangle in Tp to the halfspace above hp, and project each (nonempty) clipped triangle centrally from p onto πp. The resulting set of projected triangles has the following structure. Each point u ∈ S that lies above hp is mapped to a point 4.2 k-Sets and Triangles in IR3 45

∗ u ∈ πp. Each triangle puv in Tp for which both u and v lie above hp is mapped to the ∗ ∗ segment u v , and each triangle puv in Tp for which u lies above hp but v lies below hp ∗ is mapped to a ray emanating from u . Triangles puv in Tp for which both of u and v lie below hp are excluded from the analysis. Let Gp denote this geometric graph drawn on πp (strictly speaking, Gp is not a geometric graph in the sense of [Pac91], because of ∗ the inﬁnite rays that it contains), and let Sp be its set of vertices, the projected images of points of S above hp. We refer to both the bounded edges and the rays as edges of Gp. Notice that a crossing pair of edges in Gp corresponds to a crossing pair of triangles in Tp. We do not necessarily get all crossing pairs of triangles in Tp this way, nevertheless Lemma 4.3 bounds the total number of edge crossings in the graphs Gp. Let ep and rp be the number of (bounded or unbounded) edges and the number of rays in Gp, respectively. In the following lemma we ﬁnd the average of these numbers. P Lemma 4.5. (a) p∈S ep = 2t; P (b) p∈S rp = t. Proof. Consider any triangle ∆ in T and let the vertices of ∆ in ascending order of their z-coordinates be p, q, and r. The triangle ∆ contributes a bounded edge to Gp since q and r are both above hp. ∆ contributes a ray to Gq since r is above hq but p is below it. Finally, ∆ does not contribute to Gr since both p and q are below hr. Each triangle in T contributes two to the sum in (a) and one to the sum in (b), thus proving the lemma.

We next observe that Gp has the following antipodality property, which is an immediate corollary of the antipodality property of Lemma 4.4.

∗ ∗ ∗ Lemma 4.6. Let u ∈ Sp and let us sort the edges of Gp incident to u in the angular ∗ order around u . For any two consecutive elements e1 and e2 of this cyclic order there ∗ ∗ is a unique ‘antipodal’ edge e3 in Gp incident to u , namely, one that extends from u into the wedge that is antipodal to the wedge formed between e1 and e2.

∗ Proof. The edges e in Gp incident to u are in 1-1 correspondence with the triangles in T that are incident to both p and u. (Here u∗ is the projected image of the point u ∈ S.) Our lemma follows from Lemma 4.4 since the cyclic ordering of these edges coincides with the cyclic ordering of the triangles around the line pu and antipodality for edges corresponds to antipodality of triangles.

We use the antipodality established above to decompose the edges of each Gp into a collection of x-monotone convex chains, in a manner similar to that in [Dey98]. Notice that our assumption on the coordinate system implies that no edge of Gp is parallel to the y-axis, and thus we can distinguish between left and right endpoints of edges. For deﬁning the chains we describe how to continue a chain to the right past ∗ an edge e of Gp. We extend e to the right past its right endpoint q and turn the extended segment about q∗ counterclockwise (looking from above) until we encounter 0 ∗ the ﬁrst edge e in Gp incident to q and extending from it to the right. The chain 46 k-Sets in IR3

containing e continues through e0. If e is a ray having no right endpoint or if there is no such e0 as required, then e is the rightmost edge in its chain. A chain is extended to the left in a fully symmetric manner, replacing ‘right’ by ‘left’ and ‘counterclockwise’ by ‘clockwise’. The proof of Lemma 4.7 below implies that these right-extension and left-extension rules are consistent with each other. See Figure 4.2 for an illustration of the decomposition of Gp into chains.

c f

d a g

Figure 4.2: An illustration of the graph Gp. One convex chain is drawn as dashed and one as dotted. The remaining chains are: (−∞, a, c, +∞), (c, g, +∞), (d, e), (a, b, g), (−∞, b). Here −∞/ + ∞ means that the chain starts/stops on a ray. The (−∞, a, c, +∞) chain contains the lower ray ending at a.

Lemma 4.7. (a) Each edge of Gp appears in a unique chain.

∗ (b) A single chain terminates at any given vertex of Sp (either on its right side or on its left side).

Proof. For (a) it suffices to show that no two different edges of Gp with a common right endpoint can have the same right neighbor in their respective chains. Consider a ∗ ∗ ∗ vertex q ∈ Sp and let the edges in Gp extending from q to the left in counterclockwise ∗ angular order be e1, . . . , ek. Using Lemma 4.6 we find a unique edge fi incident to q in the wedge antipodal to eiei+1 for each of the values i = 1, . . . , k − 1. Note that since ∗ the wedges are pairwise openly disjoint, the edges fi are distinct, and extend from q to the right. Our construction guarantees that the chain containing ei continues through fi for i = 1, . . . , k − 1 and the chain through ek does not continue through any of the edges fi. 4.2 k-Sets and Triangles in IR3 47

∗ For (b), notice that if there are no edges incident to q other than the edges ei and ∗ fi then the chain containing ek terminates at q (and this is the only chain terminating ∗ there). If, however, there are more edges of Gp incident to q , then (again by Lemma 4.6) there are exactly two more edges, both extending from q∗ to the right, and the chain containing ek extends through one of them, while the other edge represents a chain that terminates (on its left) at q∗. We remark here that the above arguments also prove that the dual deﬁnition (of continuing chains to the left) results in the same set of convex chains. For (c) notice that each chain contains at most two rays.

The following lemma implies that for typical values of ep and rp (which are both 2 Θ(t/n)) and for t ≥ 100n , a positive fraction of all pairs of edges in Gp are crossing. 3 2 This is a substantial improvement over the Ω(|Tp| /n ) bound on the crossing number of the graph obtained by projecting Tp centrally from p to a sphere around p (see, e.g., [PA95, Theorem 14.12]). (Notice that Gp is the central projection from p onto πp of the portion of this spherical graph that lies in the upper hemisphere.) Using this weaker bound instead (and comparing it with the upper bound of Lemma 4.3), would yield a simple alternative proof of the known result [DE94] that a set of n points in 3-space has O(n8/3) k-sets, for any ﬁxed k. See also Corollary 3.4, and the discussion given in the Introduction..

2 Lemma 4.8. The number of edge-crossings in Gp is at least rp/8 − 3epn.

Proof: We call a pair of chains C1,C2 crossing if there exist edges e1 ∈ C1, e2 ∈ C2 that cross each other (in their relative interiors). That is, pairs of chains “crossing” at a vertex do not count. In view of Lemma 4.7(a), it suffices to obtain a lower bound for the number of pairs of chains that cross each other. Instead, let us derive an upper bound for the number of non-crossing pairs of chains. Let C1,C2 be a non-crossing pair of chains. Then either (a) C1 and C2 are disjoint, or (b) C1 and C2 meet at a vertex. We assume that both C1 and C2 start and end on rays of Gp. The total number of pairs of chains that violate this assumption is at most cpn, as follows from Lemma 4.7(b). Suppose that C1 and C2 are disjoint, in which case one of the chains, say C1, lies fully above C2 (in the y-direction). Take any edge e2 of C2, and let `1 be the line tangent to C1 and parallel to e2. (The line `1 exists because C1 lies above C2 and C2 lies above the line containing e2.) Let p1 be a vertex of C1 incident to `1; see Figure 4.3. The pair (p1, e2) determines the pair (C1,C2). Indeed, the edge e2 identifies the chain C2 uniquely, by Lemma 4.7(a). The pair (e2, p1) determine the tangent line `1, and the construction of the chains is easily seen to imply that p1 and `1 uniquely identify C1. Hence, the number of disjoint pairs of chains is at most epn. Suppose next that C1 and C2 meet at a vertex. Let e1 ∈ C1 and e2 ∈ C2 be edges of the chains with a common right endpoint. Clearly e1 and e2 determine C1 and C2. Here e1 is one of the ep edges of Gp and e2 is one of the at most n edges in Gp incident to the right endpoint of e1. (Here we use the fact that the maximum degree of Gp is bounded by n, since at most n triangles in T are incident to a fixed pair of points of S.) Hence, the number of pairs of chains having a common vertex is at most epn. 48 k-Sets in IR3

p1 `1

Figure 4.3: A pair (C1,C2) of non-crossing chains is uniquely determined by the pair (p1, e2)

¡ ¢ cp We thus have at least 2 − cpn − 2epn crossing pairs of edges in Gp which is, by Lemma 4.7(c) and by the trivial estimate cp ≤ ep, at least the claimed number 2 rp/8 − 3epn. 2 We ﬁnish the proof by comparing the upper bound in Lemma 4.3 and the lower bound in Lemma 4.8 for the number X of crossing pairs of triangles in T with a common vertex. We have X 4 2 2 3n /8 ≥ X ≥ (rp/8 − 3epn) ≥ t /(8n) − 6tn, p∈S

where the last inequality follows from Lemma 4.5. We thus have t2 ≤ 3n5 + 48tn2, which implies that t = O(n5/2). This, and the observations in Section 4.2.1 (a) and (c), complete the proof of The- orem 4.1. 2

4.2.3 Proof of Theorem 3.7 Recall that, in the setup of Theorem 3.6, we want to show that number X of crossing pairs of facets of the polytopes in P satisfies X = Ω(t2/n). Let T denote the set of all (triangular) facets of the polytopes in P. We use the same notations Gp, rp, ep , and follow closely the analysis described above. The main difference is that the convex chains in Gp are naturally induced by the polytopes in P that are incident to p. Specifically, for each point p and for each polytope P that contains p as a vertex, the set of all facets of P that are incident to p and p is not their highest vertex show up in Gp as a convex chain which starts and ends at a ray if p is not the lowest vertex of P . OtherwiseP the chain is a closed convex polygon. The total number of chains is thus k + p rp/2 (each polytope induces one closed chain). It is easily seen that Lemma 4.8 continues to hold in the new setting. Namely, we can bound the number of non-crossing chains in Gp in the same manner as in Lemma 4.8, using 4.2 k-Sets and Triangles in IR3 49

the tangency property of the polytopes in P. In more detail, the tangency property implies that a pair of non-crossing chains, as in Figure 4.3, can be uniquely charged to the corresponding pair (p1, e2). We¡ ﬁnish¢ the proof by comparing the lower bound on n 2 X with the trivial upper bound 3 · 2 · 2k = O(n k). 2

4.2.4 Open Problems (a) Our analysis is based on the upper bound O(n4) on the number of crossings derived in Lemma 4.3. However, this bound seems to be weak, because, for an edge ab connecting two points a, b of S, we want to count the number of halving triangles pcd that it crosses, with the additional constraint that pab is also a halving triangle. In our derivation we do not exploit this constraint at all, so the first open problem is whether this bound can be improved, taking into account this constraint. (b) We conjecture that the following holds: Given a set S of n points in 3-space in general position and an arbitrary set T of t triangles spanned by S, there exists a line that crosses Ω(t2/n3) triangles of T . This bound is significantly larger than the bound Ω(t3/n6) of [DE94] and it would yield a trivial proof of Theorem 4.1 (using Lovász’ Lemma). We are not aware of any construction that contradicts this conjectured bound. This bound is best possible, for t = Ω(n2), which can be shown by a simple construction. (c) An even stronger conjecture is the following: Given a set S of n points in the plane in general position and an arbitrary set T of t triangles spanned by S, there exists a point that lies in Ω(t2/n3) triangles of T . As already mentioned, the best known lower bound, due to [ACE+91], is Ω(t3/(n6 log5 n)). Again, the conjectured bound is best possible for t = Ω(n2). (Note that if (c) is true then the following strengthening of (b) also holds: Given S and T as in (b), then for any direction u there exists a line parallel to u that crosses Ω(t2/n3) triangles of T .) This variant of point selection appears to be much harder than those studied in Chapter 2. (d) Finally, can the technique used in this section be extended to higher dimensions? A main difficulty in such an extension is that, already in four dimensions, the fact that two halving simplices (even with some common vertices) cross each other does not necessarily imply that an edge of one of them crosses the relative interior of the other. This precludes an immediate extension of Lemma 4.3 to four dimensions. 50 k-Sets in IR3 Chapter 5

Generalized Geometric Graphs and Pseudo-line Arrangements

5.1 Introduction

The results of this chapter have been obtained with Micha Sharir and appear in [SS03a]. Let Γ be a collection of n (x-monotone) pseudo-lines in the plane, which we define to be graphs of continuous totally-defined functions, each pair of which intersect in exactly one point, and the curves cross each other at that point. In what follows we assume general position of the pseudo-lines, meaning that no three pseudo-lines pass through a common point, and that the x-coordinates of any two intersection points of the pseudo-lines are distinct. Let E be a subset of the vertices of the arrangement A(Γ). E induces a graph G = (Γ,E) on Γ (in what follows, we refer to such a graph as a pseudo-line graph). For each pair (γ, γ0) of distinct pseudo-lines in Γ, we denote by W (γ, γ0) the double wedge formed between γ and γ0, that is, the (open) region consisting of all points that lie above one of these pseudo-lines and below the other. We also denote by W c(γ, γ0) the complementary (open) double wedge, consisting of all points that lie either above both curves or below both curves. Definition 5.1. We say that two edges (γ, γ0) and (δ, δ0) of G form a diamond if the point γ ∩ γ0 is contained in the double wedge W (δ, δ0), and the point δ ∩ δ0 is contained in the double wedge W (γ, γ0). See Figure 5.1(i). Definition 5.2. We say that two edges (γ, γ0) and (δ, δ0) of G form an anti-diamond if the point γ ∩ γ0 is not contained in the double wedge W (δ, δ0), and the point δ ∩ δ0 is not contained in the double wedge W (γ, γ0); that is, γ ∩ γ0 lies in W c(δ, δ0) and δ ∩ δ0 lies in W c(γ, γ0). See Figure 5.1(ii).

Deﬁnition 5.3. (a) A collection S of x-monotone bounded Jordan arcs is called a collection of pseudo-segments if each pair of arcs of S intersect in at most one point, where they cross each other. (b) S is called a collection of extendible pseudo-segments if there exists a set Γ of pseudo-lines, with |Γ| = |S|, such that each s ∈ S is contained in a unique pseudo-line of Γ. 52 Generalized Geometric Graphs and Pseudo-line Arrangements

(ii) (i)

Figure 5.1: (i) A diamond. (ii) An anti-diamond.

See [Cha03] for more details concerning extendible pseudo-segments. Note that not every collection of pseudo-segments is extendible, as shown by the simple example depicted in Figure 5.2.

Figure 5.2: Three pseudo-segments that are not extendible.

Deﬁnition 5.4. (a) A drawing of a graph G = (Γ,E) in the plane is a mapping that maps each vertex v ∈ Γ to a point in the plane, and each edge e = uv of E to a Jordan arc connecting the images of u and v, such that no three arcs are concurrent at their relative interiors, and the relative interior of no arc is incident to a vertex. Such a drawing is called a topological graph; see, e.g., [Pac99]. (b) If the images of the edges of E form a family of extendible pseudo-segments then we refer to the drawing of G as an (x-monotone) generalized geometric graph. In the literature (see, e.g., [Pac99]), the term geometric graphs is usually reserved to drawings of graphs where the edges are drawn as straight line segments. In this chapter we prove the following results.

Duality between pseudo-line graphs and generalized geometric graphs. The ﬁrst main result of this chapter establishes an equivalence between pseudo-line graphs and generalized geometric graphs. We ﬁrst derive the following weaker result, which has an easy and self-contained proof. 5.1 Introduction 53

Theorem 5.5. Let Γ and G be as above. Then there is a drawing of G in the plane such that two edges e and e0 of G form a diamond if and only if their corresponding drawings cross each other an odd number of times. After the original discovery of the results presented in this chapter, Agarwal and Sharir [AS02] established a duality transformation in arrangements of pseudo-lines, which has several useful properties and other applications. Using their technique, we derive the following stronger result: Theorem 5.6. (a) Let Γ and G be as above. Then there is a drawing of G in the plane, with the edges constituting a family of extendible pseudo-segments, such that, for any two edges e, e0 of G, e and e0 form a diamond if and only if their corresponding drawings cross each other. (b) Conversely, for any graph G = (V,E) drawn in the plane with its edges constituting a family of extendible pseudo-segments, there exists a family Γ of pseudo-lines and a 1-1 mapping ϕ from V onto Γ, so that each edge uv ∈ E is mapped to the vertex ϕ(u)∩ϕ(v) of A(Γ), such that two edges in E cross each other if and only if their images are two vertices of A(Γ) that form a diamond.

Applications. As an immediate corollary of Theorem 5.6 (which can also be derived from Theorem 5.5, using the Hanani-Tutte theorem (see [Han34], [Tut70]) that any graph drawn in the plane such that every pair of edges on four distinct vertices cross an even number of times, is planar), we obtain Theorem 5.7. Let Γ and G be as above. If G is diamond-free then G is planar and thus |E| ≤ 3n − 6. Theorem 5.7 has been proven by Tamaki and Tokuyama [TT97], using a considerably more involved argument. This was the underlying theorem that enabled them to extend Dey’s improved bound of O(n4/3) on the complexity of a single level in an arrangement of lines [Dey98], to arrangements of pseudo-lines. Note that the planarity of G is obvious for the case of lines: If we dualize the given lines into points, using the duality y = ax + b 7→ (a, b) and (c, d) 7→ y = −cx + d, presented in [Ede87] and mentioned in Section 3.2, and map each edge (γ, γ0) of G to the straight segment connecting the points dual to γ and γ0, we obtain a crossing-free drawing of G. Hence, Theorem 5.7 is a natural (though harder to derive) extension of this property to the case of pseudo-lines. We note also that the converse statement of Theorem 5.7 is trivial: Every planar graph can be realized as a diamond-free pseudo-line graph (in fact, in an arrangement of lines): We draw the graph as a straight-edge graph (which is always possible [F´ar48]), and apply the inverse duality to the one just mentioned. In more generality, we can take any theorem that involves generalized geometric graphs (whose edges are extendible pseudo-segments), and that studies the crossing pattern of these edges, and ‘transport’ it into the domain of pseudo-line graphs. As an example of this, we have: Theorem 5.8. Let Γ and G be as above. (i) If G contains no three edges which form pairwise diamonds then G is quasi-planar (in the terminology of [AAP+97]; see below), 54 Generalized Geometric Graphs and Pseudo-line Arrangements

and thus its size is O(n). (ii) If G contains no k edges which form pairwise diamonds (for any ﬁxed k ≥ 4) then the size of G is O(n log n) (with the constant of proportionality depending on k).

In its appropriate reformulation in the context of generalized geometric graphs, Theorem 5.8(i) corresponds to a result of Agarwal et al. [AAP+97] on quasi-planar graphs. A quasi-planar (respectively, k-quasi-planar) graph is a graph that can be drawn in the plane such that no three (respectively, k) of its edges are pairwise crossing. It was shown in [AAP+97] that the size of a quasi-planar graph is O(n). Recent extensions and a simpliﬁed proof are presented in [PRT03]. This result was extended by Valtr [Val98a] to the case k ≥ 4 and our Theorem 5.8(ii) is a similar interpretation of Valtr’s bound in the context of pseudo-line graphs. Our reformulations are valid, for both parts of the theorem, since both the results of [AAP+97, Val98b] hold for graphs whose edges are extendible pseudo-segments.

Deﬁnition 5.9. A thrackle is a drawing of a graph in the plane so that every pair of edges either have a common endpoint and are otherwise disjoint, or else they intersect in exactly one point where they cross each other. The notion of a thrackle is due to Conway, who conjectured that the number of edges in a thrackle is at most the number of vertices. The study of thrackles has drawn much attention. Two recent papers [LPS97] and [CN00] obtain linear bounds for the size of a general thrackle, but with constants of proportionality that are greater than 1. The conjecture is known to hold for straight-edge thrackles [Pac99], and, in Section 5.6, we extend the result, and the proof, to the case of graphs whose edges are extendible pseudo-segments. That is, we show:

Theorem 5.10. Let Γ and G be as above. If every pair of edges connecting four distinct vertices (that is, curves of Γ) in G form a diamond, then the size of G is at most n. Our proof extends ideas used by Perles in the proof for the straight-edge case.

Pseudo-line graphs without anti-diamonds. We now turn to study pseudo-line graphs that do not have any anti-diamond. We show:

Theorem 5.11. Let Γ and G be as above. If G is anti-diamond-free then |E| ≤ 2n − 2. Theorem 5.11 is an extension, to the case of pseudo-lines, of a (dual version of a) theorem of Katchalski and Last [KL98], reﬁned by Valtr [Val98b], both solving a problem posed by Kupitz. The theorem states that a straight-edge graph on n points in the plane, which does not have any pair of parallel edges, has at most 2n − 2 edges and that this bound can be attained. A pair of segments e, e0 is said to be parallel (or avoiding) if the line containing e does not cross e0 and the line containing e0 does not cross e. (For straight edges, this is equivalent to the condition that e and e0 are in convex position.) The dual version of a pair of parallel edges is a pair of vertices in a line arrangement that form an anti-diamond. Hence, Theorem 5.11 is indeed an extension of the result of [KL98, Val98b] to the case of pseudo-lines. The proof, for the 5.1 Introduction 55

case of straight-edge graphs, has been recently simpliﬁed by Valtr [Val99]. Our proof, obtained independently, can be viewed as an extension of this new proof to the case of pseudo-lines. Note that Theorem 5.11 is not directly obtainable from [KL98, Val98b, Val99], (a) because Theorem 5.6 does not cater to anti-diamonds, and (b) because the analysis of [KL98, Val98b, Val99] only applies to straight-edge graphs.

The complexity of the k-level in an arrangement of pseudo-lines. We provide a simpler proof of the following result:

Theorem 5.12. The maximum complexity of the k-level in an arrangement of n pseudo- lines is O(nk1/3).

The k-level in the arrangement of a set Γ of n pseudo-lines is the (closure of) the set of all points that lie on curves of Γ and have exactly k other curves passing below them. For the case of lines, this is the dual structure of k-sets and k-edges, as studied in Chapter 4. This is a central structure in arrangements, with a long and rich history, and with many applications, both in discrete and in computational geometry. It follows from the breakthrough result of Dey [Dey98] that the complexity (number of vertices) of the k-level in an arrangement of n lines is O(nk1/3) (and T´oth’snear-linear construction [T´ot01]yields the best known lower bound). This upper bound was extended to the case of pseudo-lines by Tamaki and Tokuyama [TT97], which is based on a fairly complicated proof of Theorem 5.7. We present a much simpler proof (than both proofs in [Dey98] and [TT97]) for the general case of pseudo-lines.

Incidences and many faces in pseudo-line arrangements. Finally, as an application of Theorem 5.7, we provide yet another simple proof of the following well-known result:

Theorem 5.13. (a) The maximum number of incidences between m distinct points and n distinct pseudo-lines is Θ(m2/3n2/3 + m + n).

(b) The maximum number of edges bounding m distinct faces in an arrangement of n pseudo-lines is Θ(m2/3n2/3 + n).

The proof is in some sense ‘dual’ to the proofs based on Székely’s technique [DP98, Szé97]. The proof of Theorem 5.13(b) can be extended to yield the following result, recently obtained in [AAS03], where it has been proved using the dual approach, based on Székely’s technique.

Theorem 5.14. The maximum number of edges bounding m distinct faces in an arrangement of n extendible pseudo-segments is Θ((m + n)2/3n2/3). 56 Generalized Geometric Graphs and Pseudo-line Arrangements

5.2 Drawing Pseudo-line Graphs

In this section we prove Theorems 5.5 and 5.6. Both proofs use the same drawing rule for realizing pseudo-line graphs as generalized geometric graphs. The diﬀerence is that the stronger properties of Theorem 5.6 follow from the more sophisticated machinery of point-pseudo-line duality, developed in [AS02]. On the other hand, the proof of Theorem 5.5 is simple and self-contained. Proof of Theorem 5.5: Let ` be a vertical line such that all vertices of the arrangement A(Γ) lie to the right of `. Enumerate the pseudo-lines of Γ as γ1, . . . , γn, ordered in increasing y-coordinates of the intersection points pi = ` ∩ γi. We construct a drawing of G in the plane, using the set P = {p1, . . . , pn} as the set of vertices. For each edge (γi, γj) ∈ E, we connect the points pi and pj by a y-monotone curve ei,j, which is drawn arbitrarily, except for the following rule that speciﬁes how ei,j “navigates” around intermediate vertices pk for k between i and j.

Drawing rule: If the pseudo-line γk passes above γi ∩ γj then ei,j passes to the left of pk, otherwise ei,j passes to the right of pk.

This drawing rule is a variant of a rule recently proposed in [ANP+03] for drawing, and proving the planarity, of another kind of graphs related to arrangements of pseudo- circles or pseudo-parabolas. Note that this rule does not uniquely deﬁne the drawing. We need the following technical lemma:

Lemma 5.15. Let x1 < x2 < x3 < x4 be four real numbers. (i) Let e1,4 and e2,3 be two x-monotone Jordan arcs with endpoints at (x1, 0), (x4, 0) and (x2, 0), (x3, 0), respectively, so that e1,4 does not pass through (x2, 0) or through (x3, 0). Then e1,4 and e2,3 cross an odd number of times if and only if e1,4 passes around the points (x2, 0) and (x3, 0) on diﬀerent sides. See Figure 5.3(a). (ii) Let e1,3 and e2,4 be two x-monotone Jordan arcs with endpoints at (x1, 0), (x3, 0) and (x2, 0), (x4, 0), respectively, so that e1,3 does not pass through (x2, 0) and e2,4 does not pass through (x3, 0). Then e1,3 and e2,4 cross an odd number of times if and only if e1,3 passes below (x2, 0) and e2,4 passes below (x3, 0), or e1,3 passes above (x2, 0) and e2,4 passes above (x3, 0). See Figure 5.3(b).

Proof. In case (i), let f1 and f2 be the two real (partially deﬁned) continuous functions whose graphs are e1,4 and e2,3, respectively. Similarly, for case (ii), let f1 and f2 be the functions whose graphs are e1,3 and e2,4, respectively. Consider the function g = f1 − f2 over the interval [x2, x3]. By the intermediate- value theorem, g(x2) and g(x3) have diﬀerent signs if and only if g vanishes an odd number of times over this interval. This completes the proof of the Lemma.

Let e1 = ex,y, e2 = ez,w be the drawings of two distinct edges in G that do not share a vertex. We consider two possible cases: Case (i): The intervals pxpy and pzpw (on the line `) are nested. That is, their endpoints are ordered, up to symmetry, as pz, px, py, pw in y-increasing order along the 5.2 Drawing Pseudo-line Graphs 57

x4 x1 x2 x3

(a)

x x1 x2 x3 4 (b)

Figure 5.3: Two instances where a pair of drawn edges have an odd number of crossings. (a) The nested case. (b) The interleaving case.

line `. By Lemma 5.15, e1 and e2 cross an odd number of times if and only if e2 passes around the points px and py on diﬀerent sides. On the other hand, it is easily checked that the drawing rule implies that e1 and e2 form a diamond in G if and only if e2 passes around the points px and py on diﬀerent sides. Hence, in this case we have that e1 and e2 form a diamond if and only if they cross an odd number of times. See Figure 5.4 for an illustration. Case (ii): The intervals pxpy and pzpw ‘interleave’, so that the y-order of the endpoints of e1 and e2 is, up to symmetry, px, pz, py, pw. By Lemma 5.15, e1 and e2 cross an odd number of times if and only if e1 passes around the point pz on the same side that e2 passes around the point py. On the other hand, the drawing rule for e1 and e2 easily implies that e1 and e2 form a diamond if and only if e1 passes around the point pz on the same side that e2 passes around the point py. See Figure 5.5 for an illustration.

γz

γx py

γy px

p z γw

Figure 5.4: A diamond, and the resulting crossing, in the case that the segments pxpy and pzpw are nested. 58 Generalized Geometric Graphs and Pseudo-line Arrangements

pw γx

γ py z

pz γy

γw px

Figure 5.5: A diamond, and the resulting crossing, in the case that the segments pxpy and pzpw are interleaved.

It is also easily checked that, in the case where the intervals pxpy and pzpw are disjoint, the edges e1 and e2 do not form a diamond, nor can their drawings intersect each other. This completes the proof of Theorem 5.5. 2 Proof of Theorem 5.6: The drawing rule used in the proof of Theorem 5.5 is in fact a special case of the duality transform between points and (x-monotone) pseudo-lines, as obtained recently by Agarwal and Sharir [AS02]. Speciﬁcally, we apply this result to Γ and to the set G of the given vertices of A(Γ). The duality of [AS02] maps the points of G to a set G∗ of x-monotone pseudo-lines, and maps the pseudo-lines of Γ to a set Γ∗ of points, so that a point v ∈ G lies on (resp., above, below) a curve γ ∈ Γ if and only if the dual pseudo-line v∗ passes through (resp., above, below) the dual point γ∗. Finally, in the transformation of [AS02], the points of Γ∗ are arranged along the x-axis in the same order as that of the intercepts of these curves with the vertical line ` deﬁned above. We apply this transformation to Γ and G. In addition, for each vertex v ∈ G, ∗ incident to two pseudo-lines γ1, γ2 ∈ Γ, we trim the dual pseudo-line v to its portion ∗ ∗ between the points γ1 , γ2 . This yields a plane drawing of the graph G, whose edges form a collection of extendible pseudo-segments. The drawing has the following main property:

Lemma 5.16. Let v = γ1 ∩ γ2 and w = γ3 ∩ γ4 be two vertices in G, deﬁned by four distinct curves. Then v and w form a diamond if and only if the corresponding edges of the drawing cross each other.

Proof. The proof is an easy consequence of the proof of Theorem 5.5 given above. In fact, it suﬃces to show that the duality transformation of [AS02] obeys the drawing rule used in the above proof, with an appropriate rotation of the plane by 90 degrees. So let γi, γj, γk ∈ Γ such that γk passes above (resp., below) γi ∩ γj, and such that γk meets the vertical line ` at a point between γi ∩ ` and γj ∩ `. Our drawing rule then requires 5.3 The Complexity of a k-Level in Pseudo-line Arrangements 59

that the edge pipj pass to the left (resp., to the right) of pk. On the other hand, the ∗ ∗ duality transform, preserving the above/below relationship, makes the edge γi γj pass ∗ below (resp., above) γk. Hence the two rules coincide, after an appropriate rotation of the plane, and the lemma is now an easy consequence of the preceding analysis.

Lemma 5.16 thus implies Theorem 5.6(a). To prove the converse part (b), let G = (V,E) be a graph drawn in the plane so that its edges form a collection of extendible pseudo-segments, and let Λ denote the family of pseudo-lines containing the edges of E. Apply the point-pseudo-line duality transform of [AS02] to V and Λ. We obtain a family V ∗ of pseudo-lines and a set Λ∗ of points, so that the incidence and the above/below relations between V and Λ are both preserved. It is now routine to verify, as in the case of point-line duality, that two edges u1v1 and u2v2 of E cross each other if and only if ∗ ∗ ∗ ∗ ∗ the corresponding vertices u1 ∩ v1, u2 ∩ v2 of A(V ) form a diamond. This completes the proof of Theorem 5.6. 2 The immediate implications of these results, namely Theorems 5.7 and 5.8, follow as well, as discussed in the introduction.

5.3 The Complexity of a k-Level in Pseudo-line Ar- rangements

In this section we provide a simpler proof of the well-known O(nk1/3) upper bound on the complexity of the k-level in an arrangement of n pseudo-lines (see [Dey98, TT97]). Let Γ be the given collection of n pseudo-lines, and let E be the set of vertices of the k-level, where k is a ﬁxed constant (0 ≤ k ≤ n − 2). Theorem 5.6 and the probabilistic argument that we have repeatedly used in the preceding chapters allow us to derive the following extension of the Crossing Lemma of [ACNS82, Lei83]; We omit the obvious proof here.

Lemma 5.17 (Extended Crossing Lemma). Let G(Γ,E) be a pseudo-line graph on n pseudo-lines, with |E| ≥ 4n. Then G has Ω(|E|3/n2) diamonds.

Remark: In the restricted case where Γ is a collection of real lines, Lemma 5.17 is a dual version of the Crossing Lemma of [ACNS82, Lei83]. Dey [Dey98] has shown that the number of diamonds in G is at most the total number of vertices in the arrangement A(Γ) that lie at levels less than k. It is well known (see e.g. [CS89]) that the overall complexity of the first k levels in an arrangement of n lines or pseudo-lines is O(nk). Hence, this fact, combined with the lower bound discussed above, yield the O(nk1/3) upper bound on the complexity of the k-level. We provide here an alternative simpler proof that the number of diamonds in G is at most k(n − k − 1), without using the bound on the complexity of the first k levels. We use the fact that the vertices of the k-level can be grouped into vertex sets of k ‘concave’ vertex-disjoint chains c1, . . . , ck. Each such chain ci is an x-monotone (connected) path that is contained in the union of the pseudo-lines of Γ, such that all the vertices of ci are at level k. Informally, as we traverse ci from left to right, 60 Generalized Geometric Graphs and Pseudo-line Arrangements whenever we reach a vertex of A(Γ), we can either continue along the pseudo-line we are currently on, or make a right (i.e., downward) turn onto the other pseudo-line, but we cannot make a left (upward) turn; in case the pseudo-lines are real lines, ci is indeed a concave polygonal chain. (In this case, the dual structure of such a chain is a convex polygonal chain connecting some of the dual points and consisting of k-edges, similar to the chains used in the analysis in chapters 3 and 4.) The simple construction of these chains is described in [AACS98]: Each chain starts on each of the k lowest pseudo-lines at x = −∞, and makes (right) turns at every vertex of the k-level that it encounters, and only at those vertices. In a symmetric manner, we can group the vertices of the k-level into n − k − 1 ‘convex’ vertex-disjoint chains, by starting the chains along the n − k − 1 highest pseudo-lines at x = −∞, and by making left turns at every vertex of the k-level that we encounter, and only at those vertices. Let (p, p0) be a diamond, where p and p0 are vertices at level k. Assume without loss of generality that p lies 0 to the left of p . Let c1 be the unique concave chain that contains p and let c2 be the unique convex chain that contains p0. See Figure 5.6 for an illustration. For a given vertex v in A(Γ), let Wr(v) (resp. Wl(v)) denote the (interior of the) right (resp., left) wedge of the double-wedge formed by the two pseudo-lines defining v. Consider the right wedges of vertices of c1. It is easy to see (from the ‘concavity’ of c1) that those wedges are pairwise disjoint (see also [AACS98]). A similar argument holds for the left 0 0 wedges of the vertices of c2. Since p ∈ Wr(p) and p ∈ Wl(p ), it follows that c2 does not meet the lower edge of Wr(p), but meets the upper edge of this wedge. This can happen for at most one vertex of c1, because of the disjointness of the right wedges of its vertices. Hence p is uniquely determined from the pair (c1, c2), and, symmetrically, this also holds for p0.

p p0

Figure 5.6: An arrangement of 5 pseudo-lines and a diamond formed by two vertices 0 (p, p ) at level 1, the corresponding concave chain c1 containing p, and the convex chain 0 c2 containing p .

Thus the number of diamonds in the k-level is at most the number of pairs (c1, c2) of a concave chain and a convex chain; that is, at most k(n − k − 1). This completes 5.4 Yet Another Proof for Incidences and Many Faces in Pseudo-line Arrangements 61 the proof of Theorem 5.12. 2

5.4 Yet Another Proof for Incidences and Many Faces in Pseudo-line Arrangements

In this section we provide yet another proof of the well-known (worst-case tight) bounds given in Theorem 5.13. We will prove only part (b) of the theorem; part (a) can then be obtained by a simple and known reduction (see, e.g., [CEG+90]); alternatively, it can be obtained by a straightforward modiﬁcation of the proof of (b), given below.

Let Γ be the given collection of n pseudo-lines, and let f1, . . . , fm be the m given faces of the arrangement A(Γ). Let E denote the set of all vertices of these faces, excluding the leftmost and rightmost vertex, if any, of each face. Since every bounded face has at least one vertex that is not leftmost or rightmost, and since the number of unbounded faces is O(n), it follows that the quantity that we wish to bound is O(|E| + n). By Lemma 5.17, if |E| ≥ 4n then the graph G(Γ,E) has Ω(|E|3/n2) diamonds. Let (p, p0) be a diamond, where p is a vertex of some face f and p0 is a vertex of another face f 0. (It is easily verified that if p and p0 bound the same face then they cannot form a diamond.) Then, using the Levy Enlargement Lemma [Lev26], there exists a curve γ0 0 that passes through p and p , such that Γ ∪ {γ0} is still a family of pseudo-lines. In this 0 case γ0 must be contained in the two double wedges of p and p , and thus it avoids the 0 0 interiors of f and of f ; that is, γ0 is a ‘common tangent’ of f and f . As in the case of lines, it is easy to show that a pair of faces can have at most four common pseudo-line tangents of this kind. Hence, the number of diamonds in G cannot exceed 2m2. Putting everything together, we obtain |E| = O(m2/3n2/3 + n). 2 Remark: This proof is, in a sense, dual to that of Székely [Szé97]for incidences, or to its extension by Dey and Pach [DP98] for many faces. These former proofs interchange the roles of points and (pseudo)lines: they apply the crossing lemma to a different graph, whose vertices are the points involved in the incidences or marking points, one in each of the given faces. The proof of Theorem 5.14 is proved in a similar manner. The main difference is that the given faces need not be x-monotone, because their boundaries may contain endpoints of the given pseudo-segments. In this case two vertices of the same face may form a diamond, and the number of diamonds formed between two distinct faces may be arbitrarily large. To overcome this issue, we partition, as in [AAS03], any such face into x-monotone sub-faces, by vertical segments erected from endpoints of the pseudo- segments. The number of new sub-faces is O(m + n), and any pair of them can induce only O(1) diamonds, which can be argued exactly as in the case of pseudo-lines. The preceding arguments then yield the asserted bound. 62 Generalized Geometric Graphs and Pseudo-line Arrangements

5.5 Graphs in Pseudo-line Arrangements without Anti-Diamonds

So far, this chapter has dealt exclusively with the existence or nonexistence of diamonds in graphs in pseudo-line arrangements. We now turn to graphs in pseudo-line arrangements that do not contain any anti-diamond. Recall that the notion of an anti- diamond is an extension, to the case of pseudo-lines, of (the dual version of) a pair of edges in (straight-edge) geometric graphs that are in convex position (so-called ‘parallel’, or ‘avoiding’ edges). Using Theorem 5.6 (and the analysis in its proof), one obtains a transformation that maps an anti-diamond-free pseudo-line graph (Γ,G) to a generalized geometric graph, whose edges form a collection of extendible pseudo-segments, with the property that, for any pair e, e0 of its edges, deﬁned by four distinct vertices, either the pseudo-line containing e crosses e0 or the pseudo-line containing e0 crosses e. We present a much shorter and simpler proof of Theorem 5.11 than those of [KL98, Val98b], that applies directly in the original pseudo-line arrangement, and is similar in spirit to the recent (independently discovered) simpliﬁed proof of Valtr [Val99] for the case of straight-edge geometric graphs.

pa,b

Figure 5.7: A subsequence ··· a ··· b ··· of A to the left of pa,b and the resulting anti- diamond.

Proof of Theorem 5.11: We construct two sequences A and B whose elements belong to Γ, as follows. We sort the intersection points of the pseudo-lines of Γ that correspond to the edges of G in increasing x-order, and denote the sorted sequence 0 by P = hp1, . . . , pmi. For each element pi of P , let γi and γi be the two pseudo-lines 0 0 forming (meeting at) pi, so that γi lies below γi to the left of pi (and lies above γi to 0 the right). Then the i-th element of A is γi and the i-th element of B is γi. Lemma 5.18. The concatenated cyclic sequence C = AkB does not contain a sub-cycle of alternating symbols of the form a ··· b ··· a ··· b, for a 6= b. 5.5 Graphs in Pseudo-line Arrangements without Anti-Diamonds 63

Proof. Assume to the contrary that C does contain such a sub-cycle. Consider the point pa,b of intersection of the curves a and b. There are two cases to consider: Case (i): a lies below b to the left of pa,b. We claim that there is no subsequence a ··· b in A to the left of pab (that is, involving elements whose associated intersection points have x-coordinates smaller than that of pa,b). Indeed, if such a subsequence existed, then there would be curves a0 and b0 in Γ such that (a, a0) and (b, b0) are edges in G, a0 is above a to the left of p = a ∩ a0, b0 is above b to the left of q = b ∩ b0, and p lies to the left of q, which lies to the left of pa,b. It is easily seen that in such a case the two edges (a, a0), (b, b0) form an anti-diamond in G (see Figure 5.7), contrary to assumption. Symmetric arguments show that there is no subsequence b ··· a of A to the right of pa,b, no subsequence b ··· a of B to the left of pa,b, and no subsequence a ··· b of B to the right of pa,b. These arguments imply that A cannot contain a subsequence a ··· b ··· a, for otherwise A would have to contain either a ··· b to the left of pa,b, or b ··· a to the right of pa,b, both of which are impossible. Similarly, B cannot contain a subsequence b ··· a ··· b, for that would imply that B would have to contain either b ··· a to the left of pa,b or a ··· b to the right of pa,b, both of which are impossible. Hence, if the concatenated sequence C contains an a ··· b ··· a ··· b then, since A cannot contain an a ··· b ··· a and B cannot contain a b ··· a ··· b, the only case to consider is that A contains an a ··· b, where b is (necessarily) to the right of pa,b, and B contains an a ··· b, where a is (necessarily) to the left of pa,b. In that case, the two intersection points that correspond to the element b of A and to the element a of B in the above subsequence form an anti-diamond (see Figure 5.8), contradicting our assumption that G is anti-diamond free.

Case (ii): b lies below a to the left of pa,b. There are three sub-cases to consider. In the first sub-case, A contains an a ··· b ··· a and B contains b. Reversing the roles of a, b in the analysis of Case (i), we conclude that A does not contain b ··· a to the left of pa,b, and a ··· b to its right. Hence, there is an intersection point of G labeled a in A to the left of pa,b, and another such point labeled a in A to the right of pa,b. It is easily verified that the edge (intersection point) e of G labeled by b in B and that edge labeled by a in A that lies on the side of pa,b opposite to e form an anti-diamond (see Figure 5.9), a contradiction that rules out this sub-case. A symmetric argument excludes the case where A contains a single a and B contains b ··· a ··· b. The third possible case is that A contains an a ··· b and B also contains an a ··· b. But again the first a (that belongs to A) must be to the left of pa,b and the second b (that belongs to B) must be to the right of pa,b. In that case those two are labels of edges that form an anti-diamond; see Figure 5.10 for an illustration. This completes the proof of the lemma.

A run in C is a maximal contiguous subsequence of identically labeled elements. If we replace each run by a single element, the resulting sequence C∗ is a Davenport- Schinzel cycle of order 2 on n symbols, as follows from Lemma 5.18. Hence, the length of C∗ is at most 2n − 2 [SA95]. 64 Generalized Geometric Graphs and Pseudo-line Arrangements

pa,b

Figure 5.8: The anti-diamond arising in the second part of Case (i).

a ∈ A pa,b

b ∈ B a ∈ A b

Figure 5.9: The anti-diamond arising in the ﬁrst of Case (ii), formed by the two lower marked vertices.

Note that it is impossible to have an index 1 ≤ i ≤ |G| such that the element of A with index i is equal to the element of A with index (i + 1)(mod|G|) and the element of B with index i is equal to the element of B with index (i + 1)(mod|G|). Indeed, if these elements are a and b, respectively, then we obtain two vertices of A(Γ) (the one encoded by the i-th elements of A and B and the one encoded by the (i + 1)-st elements) that are incident to both a and b, which is impossible. In other words, for each i = 0,..., |G| − 1, a new run must begin either after the i-th element of A or after the i-th element of B (or after both). This implies that the length of C∗ is greater than or equal to |G|. Hence we have: |G| ≤ |C∗| ≤ 2n − 2. This completes the proof of Theorem 5.11. 2

5.6 Pseudo-line and Thrackles

Let G be a thrackle with n vertices, whose edges are extendible pseudo-segments. We transform G, using the pseudo-line duality, to an intersection graph in an arrangement 5.6 Pseudo-line and Thrackles 65

pa,b

Figure 5.10: The anti-diamond arising in the third of Case (ii).

of a set Γ of n pseudo-lines. The edge set of G is mapped to a subset E of vertices of A(Γ), with the property that every pair of vertices of E, not sharing a common pseudo-line, form a diamond.

Theorem 5.19. |E| ≤ n.

Proof: The proof is an extension, to the case of pseudo-line graphs (or, rather, generalized geometric graphs drawn with extendible pseudo-segments), of the beautiful and simple proof of Perles, as reviewed, e.g., in [Pac99]. Fix a pseudo-line γ ∈ Γ and consider the vertices in E ∩ γ. We say that v ∈ E ∩ γ is a right-turn (resp., left-turn) vertex with respect to γ if, to the left of v, γ lies above (resp., below) the other pseudo-line incident to v. If γ contains three vertices v1, v2, v3 ∈ E, appearing in this left-to-right order along γ, such that v1 and v3 are right-turn vertices and v2 is a left-turn vertex, then all vertices of E must lie on γ, because the intersection of the three (open) double wedges of v1, v2, v3 is empty, as is easily checked. In this case |E| ≤ n − 1 and the theorem follows. A similar argument holds when v1 and v3 are left-turn and v2 is a right-turn vertex. Hence we may assume that, for each γ ∈ Γ, the left-turn vertices of E ∩ γ are separated from the right-turn vertices of E ∩ γ along γ. For each γ ∈ Γ, we delete one vertex of E ∩ γ, as follows. If E ∩ γ consists only of left-turn vertices, or only of right-turn vertices, we delete the rightmost vertex of E ∩ γ. Otherwise, these two groups of vertices are separated along γ, and we delete the rightmost vertex of the left group. We claim that after all these deletions, E is empty. To see this, suppose to the contrary that there remains a vertex v ∈ E, incident to two pseudo-lines γ1, γ2 ∈ Γ, such that γ1 lies below γ2 to the left of v. Clearly, v is a left-turn vertex with respect to γ1, and a right-turn vertex with respect to γ2. The deletion rule implies that, initially, E ∩ γ1 contained either a left-turn vertex − v1 that lies to the right of v (in case that either all vertices of E ∩ γ1 are left-turn, or that all left-turn vertices of E ∩ γ1 are to the left of all right-turn vertices of E ∩ γ1), 66 Generalized Geometric Graphs and Pseudo-line Arrangements

+ or a right-turn vertex v1 that lies to the left of v. Similarly, E ∩ γ2 contained either a − + left-turn vertex v2 that lies to the left of v, or a right-turn vertex v2 that lies to the right of v. It is now easy to check (see Figure 5.11) that, in each of the four possible − − + − − + + + cases, the respective pair of vertices, (v1 , v2 ), (v1 , v2 ), (v1 , v2 ), or (v1 , v2 ), do not form a diamond, a contradiction that shows that, after the deletions, E is empty. Since we delete at most one vertex from each pseudo-line, it follows that |E| ≤ n. 2

γ2 − v1 − v2

v + v+ v1 2

γ1

Figure 5.11: A vertex v remaining after the deletions implies the existence of a pair of vertices that do not form a diamond. Chapter 6

Conﬂict-Free Coloring Problems

6.1 Introduction

In this chapter we study a coloring problem which we refer to as the Conflict-Free coloring (CF-coloring for short) related to frequency assignment problems in cellular networks. In its most general form, the input of the CF-coloring problem is a set system (X, S), where each S ∈ S is a subset of X. The output is a coloring χ of the elements in X that satisfies the following constraint: for every S ∈ S there exists a color i and a unique element x ∈ X, such that x ∈ S and χ(x) = i. The goal is to minimize the number of colors used by the coloring χ. CF-coloring of general set systems is not easier than the classical graph coloring problem. However, in view of our motivation, we consider set systems that arise in geometry. Some of these set systems have nice properties and can be CF-colored with a “small” number of colors. Specifically, we focus on the following two special cases of CF-coloring of a set system: CF-coloring of regions: Given a finite family S of n regions of some fixed type (such as discs, pseudo-discs, axis-parallel rectangles, etc.), what is the minimum integer k, such that one can assign a color to each region of S, using a total of at most k colors, such that the resulting coloring has the following property: For each point p ∈ ∪b∈S b there is at least one region b ∈ S that contains p in its interior, whose color is unique among all regions in S that contain p in their interior (in this case we say that p is being ‘served’ by that color). We refer to such a coloring as a conflict-free coloring of S (CF-coloring in short). CF-coloring of a range space: A given set P of n points in IRd and a set R of ranges (for example, the set of all discs in the plane) define a so-called range space (P, R). Given such a range space, what is the minimum integer k, such that one can color the points of P by k colors, so that for any r ∈ R with P ∩ r 6= ∅, there is at least one point q ∈ P ∩ r that is assigned a unique color among all colors assigned to points of P ∩ r (in this case we say that r is ‘served’ by that color). We refer to such a coloring as a conflict-free coloring of (P, R) (CF-coloring in short). The study of such problems is motivated by the problem of frequency assignment in cellular networks. Specifically, cellular networks are heterogeneous networks with two 68 Conflict-Free Coloring Problems

different types of nodes: base stations (that act as servers) and clients. The base stations are interconnected by an external fixed backbone network. Clients are connected only to base stations; connections between clients and base stations are implemented by radio links. Fixed frequencies are assigned to base stations to enable links to clients. Clients, on the other hand, continuously scan frequencies in search of a base station with good reception. The fundamental problem of frequency assignment in cellular networks is to assign frequencies to base stations so that every client, located within the receiving range of at least one station, can be served by some base station, in the sense that the client is located within the range of the station and no other station within its reception range has the same frequency. The goal is to minimize the number of assigned frequencies since the frequency spectrum is limited and costly. Suppose we are given a set of n base stations, also referred to as antennas. Assume, for simplicity, that the area covered by a single antenna is given as a disc in the plane. Namely, the location of each antenna (base station) and its radius of transmission is fixed and is given (the transmission radii of the antennas are not necessarily equal). In Section 6.3 we show that in this case, one can find an assignment of frequencies to the antennas with a total of at most O(log n) frequencies, such that each antenna is assigned exactly one of the frequencies and the resulting assignment is free of conflicts in the preceding sense. Furthermore, we show that this bound is worst-case optimal. We also show that such a coloring can be found in polynomial time. Our proof uses a dual transformation which maps discs in the plane to points in IR3 and points in the plane to planes in IR3. We then study the problem of CF-coloring the finite range space (P, R) where P is a set of n points in IR3 and R is the set of all halfspaces in IR3. In Section 6.3 we develop a general framework for CF-coloring “nice” range spaces. In particular we show that the above range space (P, R) has a CF-coloring with O(log n) colors, which can be found in polynomial time. This result, of CF-coloring n points in IR3 with respect to halfspaces, is stronger than CF-coloring of discs in the plane, because it also handles halfspaces that do not correspond to points. Specifically, the duality transformation maps points in the plane to planes in IR3, all of which are tangent to the unit paraboloid z = x2 + y2. However, our coloring is conflict-free with respect to any halfspace in IR3. The general approach used in Section 6.3 works for any range space having a certain property which we refer to as the monotonicity property. Using this approach we show that any range space (P, R), where P is a set of n points in the plane, and R is the set of all discs, admits a CF-coloring with O(log n) colors and that this bound is tight in the worst case. We also present a polynomial-time algorithm for producing such a coloring (recall that in this new version we color the points of P , whereas in the preceding problem we colored the given discs). This result generalizes to any collection R of homothetic copies of a fixed convex body. Using a similar approach, we next study the problem of CF-coloring of range spaces, where the underlying ranges are axis-parallel rectangles in the√ plane, and show that any set of n points in the plane can be CF-colored with O( n) colors with respect to axis-parallel rectangles. Using a different approach, we also obtain non-trivial upper bounds on the number of colors needed in any CF-coloring of a range space consisting of n points in IRd whose ranges 6.1 Introduction 69

are axis-parallel√ boxes.√ We also study the special case when all the given points form the regular n × n-grid and show that in this case one can color the points with O(log n) colors and that this bound is worst-case optimal. This bound holds for any dimension. Namely for any fixed d one can color the points of the d-dimensional regular n1/d × ... × n1/d grid with O(log n) colors with respect to axis-parallel boxes. In fact, we show that the constant in the big ‘O’ notation does not depend on the dimension d. We note that, without the assumption that the rectangles are axis-parallel, the problem becomes uninteresting. Indeed, any planar set P of n points in general position (i.e., no three are co-linear) needs n colors in any CF-coloring of P with respect to arbitrarily oriented rectangles. In Section 6.4 we use more involved probabilistic and geometric ideas, and obtain a general probabilistic algorithm which CF-colors any set of n “simple” regions (not necessarily convex) whose union has “low” complexity, using a “small” number of colors. (The quoted terms are interrelated, in a manner stated more precisely below.) In particular, we show that if the regions under consideration have a union of near linear complexity, then they can be CF-colored using a polylogarithmic number of colors. This holds for pseudo-discs [KLPS86], convex α-fat shapes [ES00], and (α, β)-covered objects [Efr99]. This provides the first non-trivial and near-optimal bounds for one of the problems that motivated our work. In practice, cellular antennas are directional, and the region of influence of an antenna is a circular sector with central angle of 60o. Since such sectors are fat and convex, our results thus imply that those regions have a conflict-free coloring using a polylogarithmic number of colors. In Section 6.5, we refine the results of Section 6.4, deriving better bounds for some special cases. We show that any set of n axis-parallel rectangles in the plane can be 2 CF-colored with O(log n) colors. (Note the sharp contrast with the “dual” problem√ of coloring points with respect to axis-parallel rectangles, where only a bound of O( n) is known.) We note (as in a similar observation made above) that the assumption that the rectangles be axis-parallel cannot be removed, for otherwise one can construct a set R of n rectangles in which any CF-coloring of R needs n colors. We also study the case of a set R of regions in the plane with boundaries induced by x-monotone curves, each pair of which intersect at most s times, where s is some fixed given constant. We present a polynomial-time algorithm that CF-colors R with O(βs(n) log n) colors, where βs(n) = λs(n)/n and λs(n) is the maximum length of a Davenport-Schinzel sequence of order s with n symbols (See [SA95]). Using a similar approach we show that any set of convex regions with linear union complexity and with a nonempty intersection admits a CF-coloring with O(log n) colors. Extending this technique, we also show that any set of n pairwise intersecting pseudo-discs has a CF-coloring with a total of O(log n) colors. Finally in Section 6.7 we generalize the notion of CF-coloring of range spaces and of regions to what we call k-CF-coloring. That is, in the case of coloring a range space, we say that a range is “served” if there is a color that appears in the range (at least once and) at most k times, for some fix prescribed parameter k. This notion seems to be natural and applicable in the cellular network scenario, and provides additional flexibility. A similar generalization of k-CF-coloring a set of regions is also studied. 70 Conflict-Free Coloring Problems

For example, we show that there is a range space consisting of n points√ for which any CF-coloring needs n colors but there exists a 2-CF-coloring with O( n) colors (and a k-CF-coloring with O(n1/k) colors for any ﬁxed k ≥ 2). Some of the results presented in this chapter appear in two papers. The ﬁrst paper is a joint work with Guy Even, Zvi Lotker and Dana Ron [ELRS03]. The second paper is joint work with Sariel Har-Peled [HPS03].

6.2 A General Framework

We briefly introduce some notations and tools used in this chapter. In the following, P denotes a set of n points in IRd, and R denotes a set of ranges (for example, the set of all discs in the plane). A range space S is a pair (X, R), where X is a (finite or infinite) set and R is a (finite or infinite) family of subsets of X. If A is a subset of X then ΠR(A) = {r ∩ A : r ∈ R} is the restriction of R on A. In this chapter we focus on range spaces that arise naturally in combinatorial and computational geometry. One such example is the space S = (IRd, H), where H is the set of all halfspaces in IRd. For a finite set of points P in IRd and a (finite or infinite) collection R of ranges, we abuse the notation slightly and refer to the pair (P, R) as a range space, referring to the restriction ΠR(P ) of R on P . The “Delaunay” graph G = G(P, R) is the graph whose vertex set is P and whose edges are all pairs (u, v) for which there exists a range r ∈ R such that r ∩ P = {u, v}. We denote a range realizing an edge (u, v) ∈ G by ruv. A coloring f : P → {1, . . . , k} is a conflict-free coloring of (P, R)(CF-coloring in short), if for any r ∈ R, such that P ∩ r 6= ∅, there exists a color i, for which there is a point p ∈ P ∩ r, such that f(p) = i, and no other point of P ∩ r is assigned the color i. Any range r for which this property holds (regardless of whether the coloring is conflict free) is said to be served by the coloring. We refer to the minimum number of colors needed to CF-color (P, R) as the conflict-free (or CF)-chromatic number of (P, R). d For a set R of ranges in IR , let kopt(n, R) denote the maximum number of colors needed for the given set R, over all sets of n points in IRd. A range space (P, R) is called shrinkable if for any P1 ⊂ P and for each r ∈ R with 0 0 0 |r ∩ P1| > 2 there exists a range r ∈ R such that |r ∩ P1| = 2, and r ∩ P1 ⊂ r ∩ P1. The following is a general framework for CF-coloring a shrinkable range space (P, R).

Let Li ⊂ P denote the set of points in P colored with i by Algorithm 1. We refer to Li as the i-th layer of (P, R). Lemma 6.1. The coloring of a shrinkable range space (P, R) by Algorithm 1 is a valid CF-coloring of (P, R). Proof. Consider a range r ∈ R, such that |P ∩ r| ≥ 2. Let i be the maximal color of the points in r. Let Pi ⊂ P be the set of input points at the beginning of the i’th iteration, i.e., the set just before color i has been assigned. Note that Li ⊂ Pi and Li ∩ r = Pi ∩ r (since i is the maximal color in r). Clearly, if |r ∩ Li| = 1 then r is served and we are done. 6.2 A General Framework 71

Algorithm 1 CFcolor(P, R): CF-color a set P with respect to a set of ranges R. 1: i ← 0: i denotes an unused color 2: while P 6= ∅ do 3: Increment: i ← i + 1 4: Find an independent set P 0 ⊂ P of G(P, R): We elaborate subsequently on the implementation of this step. 5: Color: f(x) ← i , ∀x ∈ P 0 6: Prune: P ← P \ P 0 7: end while

Thus, we only have to consider the case |r ∩ Li| > 1. However, by the monotonicity 0 property (applied to the subset Pi), it follows that there exists a range r such that: (i) 0 0 |r ∩ Pi| = 2, and (ii) r ∩ Pi ⊂ r ∩ Pi = r ∩ Li. 0 This means that the two points of r ∩ Li form an edge in the graph G(Pi, R). This however contradicts the fact that Li is independent in G(Pi, R), and thereby completes the proof of the lemma. Building upon Lemma 6.1, we obtain: Lemma 6.2. Let R be a set of ranges in IRd, so that for any finite set P , the range space (P, R) is shrinkable. (i) If the Delaunay graph G(P, R) contains an independent set of size at least α|P |, log n for some fixed 0 < α < 1, then kopt(n, R) ≤ log(1/(1−α)) . (ii) If G(P, R) contains an independent set of size Ω(|P |1−²), for some fixed 0 < ² < ² 1, then kopt(n, R) = O(n ). Proof. The assumption in part (i) of the lemma implies that in the i’th iteration of Algorithm 1 we color at least α|Pi| points of Pi with the color i. This means that if we log n start with a set of n points, the number of iterations is at most log(1/(1−α)) . Similarly, part (ii) of the lemma follows by observing that the number of iterations needed by Algorithm 1 is bounded by O(n²). We need the following technical definition and lemma, for subsequent sections. Definition 6.3. For a finite set V , a k-uniform hypergraph H on V is a pair of the form (V,E), where E is a set of subsets of V of size k (those are the hyperedges of H). The degree of a vertex v ∈ V is the number of sets (i.e., hyperedges) of E that contain v. A set A ⊆ V is called an independent set if no hyperedge of E is contained in A. See [Ber89] for more details concerning uniform hypergraphs. Lemma 6.4 (Turán’sTheorem). (i) Let G be a simple graph on n vertices with average degree δ. Then G contains an independent set of size Ω(n/δ).

(ii) Let H be a k-uniform hypergraph with¡ n vertices¢ and average degree δ. Then H k 1/(k−1) contains an independent set of size Ω n( δ ) . 72 Conﬂict-Free Coloring Problems

Proof. Both facts are easy exercises in graph theory (see, e.g., [AS92]); nevertheless we provide proofs for the sake of completeness. (Note also that part (i) is a special case of part (ii), with k = 2.) (i) Remove from G all vertices of degree larger than, say, 2δ. Let G0 be the resulting graph. Clearly, the number of vertices in G0 is at least n/2, and the maximum degree in G0 is at most 2δ. Thus, G0 contains an independent set of size Ω(n/δ), which is also an independent set in G. (ii) Let H = (V,E) be the given k-uniform hypergraph. Let m = |E| be the δn 0 number of edges of H. By direct counting, we have m = k . Let S be the set of vertices generated by picking each vertex of V independently with probability p, where p will be speciﬁed shortly. For each edge e ∈ E that is fully contained in S0, we remove one of its vertices from S0. Let S denote the resulting set, which is clearly independent. k k δn 1/(k−1) We have E[|S|] ≥ pn − p m = pn − p k . Setting p = (k/2δ) , we have µ ¶ µ ¶ µ ¶ µ ¶ k 1/(k−1) δn k k/(k−1) k 1/(k−1) 1 E[|S|] ≥ n − = n 1 − . 2δ k 2δ 2δ 2 ¡ ¢ It follows that there exists an independent set in H of size Ω n(k/δ)1/(k−1) .

6.3 CF-Coloring of Range Spaces

In this Section we apply the general framework of Section 6.2 to several concrete geometric range spaces.

6.3.1 Coloring Points in the Plane with Respect to Discs Using Lemma 6.1 and Lemma 6.2 we obtain:

Theorem 6.5. Let R be the set of all discs in the plane. Then

log2 (n + 1) ≤ kopt(n, R) ≤ log4/3 (n + 1)

Moreover, a CF-coloring of (P, R), for any set P with n points, that uses at most

log6/5 n colors, can be computed in O(n log n) time. Proof. Let P be a set of n points in the plane. First, we claim that (P, R) is shrinkable. This is an easy though technical geometric exercise. Observe that G(P, R) is the standard Delaunay graph of P . More precisely, G(P, R) coincides with the standard graph if the points of P are in general position (i.e., no four points of P are co-circular). If four points (or more) of P can be co-circular, G(P, R) may be a proper subgraph of the standard Delaunay graph; see Figure 6.1 (See also the analysis of Delaunay graphs in Chapter 2.) In either case, G(P, R), is planar (see, e.g., [dBvKOS00]), and, as such, it has an independent set of size at least n/4 (by the four-color theorem [RSST96]). It follows that in the i’th step of Algorithm 1 we may discard at least |Pi| /4 points of Pi and hence, by Lemma 6.2(i), we use a total of at most log4/3 (n + 1) colors. To make 6.3 CF-Coloring of Range Spaces 73

the algorithm more eﬃcient, we can use an O(n log n)-time greedy algorithm to ﬁnd an independent set of size at least |Pi| /6 in the i’th iteration of Algorithm 1. This is done by picking a vertex with degree at most 5, removing all its neighbors, and recursing on the remaining subgraph. (Note that there always exists a vertex with degree at most 5, since the average degree of a planar graph on n vertices is at most 6 − 12/n.)

Lemma 6.2(i) implies that this method yields a CF-coloring with at most log6/5 (n + 1) colors. The construction of this coloring takes O(n log n) time. Indeed, the i’th iteration takes O(|Pi| log |Pi|) time, both for the construction of G(Pi, R) [dBvKOS00] and for the construction of the independent set. Since the sizes of the Pi’s decrease geometrically, the claim follows. As for the lower bound, let P = {1, . . . , n} be a set of n integer points on the x-axis (sorted in increasing order). The subsets of P that are realizable by intersecting P with some disc are all discrete intervals [i, j] = {i, i + 1, . . . , j}, for 1 ≤ i ≤ j ≤ n. It is easily seen that in any CF-coloring of (P, R), log2 (n + 1) colors are necessary. We will show, by induction on k, that if k colors suﬃce for coloring (P, R) then n ≤ 2k − 1. Indeed, P = [1, n] has a unique color, call it 1. Let i be the unique point that is colored by 1. The number of colors used for any of the two subintervals, [1, i − 1] and [i + 1, n], is at most k − 1. Hence, by the induction hypothesis, i − 1 ≤ 2k−1 − 1 and n − i ≤ 2k−1 − 1. Thus n = i − 1 + n − i + 1 ≤ 2k − 1. This completes the induction step and hence the proof of the lower bound.

Figure 6.1: The Delaunay graph G(P, R) in a degenerate conﬁguration.

The lower bound of Theorem 6.5 has recently been considerably strengthened by Pach and T´oth[PT03], who have shown that any set P of n points in the plane needs Ω(log n) colors in any CF-coloring of (P, R). Remark: Theorem 6.5 also holds for any range space R consisting of all homothetic copies of a ﬁxed convex body. The proof of this fact is similar to the proof of Theorem 6.5, and is given in detail in [ELRS03].

6.3.2 Coloring Points in IR3 with Respect to Halfspaces In this subsection we study range spaces (P, R), where P ⊂ IR3 is ﬁnite and R is the set of all halfspaces in IR3. Given a plane h (not parallel to the z-axis), the positive halfspace h+ (respectively, the negative halfspace h−) is the set of all points that lie on or above (resp., on or below) h. We denote by R+ the set of all positive halfspaces 74 Conﬂict-Free Coloring Problems

in IR3. We call a point p ∈ P extreme if there exists a halfspace h+ ∈ R+ such that h+ ∩ P = {p}.

+ Theorem 6.6. kopt(n, R ) = O(log n). The bound is worst-case tight. Moreover, for any set P ⊂ IR3 of n points, a CF-coloring of P with O(log n) colors can be constructed in O(n log n) time.

Proof. We make the simplifying assumption that all points of P are extreme. If not, color all the non-extreme points of P by a unique “passive” color, and then CF-color the extreme points not using that color. Since every halfspace h+ that intersects the convex hull of P must contain an extreme point of P , it follows that this coloring of P is conflict-free. We first show that (P, R+) is a shrinkable range space. Indeed, let r be a positive halfspace such that |r ∩ P | > 2. We want to show that there exists a positive halfspace r0 such that |r0 ∩ P | = 2 and r0 ∩ P ⊂ r ∩ P . Let h denote the plane bounding r. Translate h upwards as much as possible so that every further translation upwards reduces the intersection of the positive halfspace with P to fewer than 2 points. Let h1 denote the plane parallel to h obtained by this translation, and let r1 denote the range + h1 . If r1 ∩P contains more than two points, then either r1 ∩P is contained in the plane h1 or all but one of the points in r1 ∩ P are in h1. Assume that r1 ∩ P ⊂ h1. Then r1 ∩ P is the set of vertices of a convex polygon. Choose an edge uv of that polygon and rotate h1 slightly about uv, so that the new plane h2 satisfies h2 ∩ P = {u, v}. The + range h2 satisfies the desired property. A similar argument applies if there is a single point in (r1 ∩ P ) \ h1, and the claim follows. We next observe that the Delaunay graph G(P, R+) is just the skeleton graph of the convex hull of P , which is planar and hence has an independent set of size at least |P |/4. Plugging this fact into Lemma 6.2, we have that (P, R) can be CF-colored with O(log n) colors. To see that the bound is tight, place n points on the vertical parabola y = 0, z = −x2, and apply the same analysis as in the proof of Theorem 6.5. An O(n log n) time algorithm for constructing a coloring can be obtained in much the same way as in the proof of Theorem 6.5.

6.3.3 Axis-parallel Rectangles In this subsection, we deal with the problem of conﬂict-free coloring of points in the plane, where the ranges are axis-parallel rectangles.

Theorem 6.7. For the set B2 of all axis-parallel rectangles in the plane, we have 2 √ kopt(n, B ) = O( n).

Proof. Let P be a set of n points in the plane, and let G = G(P, B2) denote the corresponding Delaunay graph. Note that the ranges that realize the edges of G can be taken to be those rectangles that have two points of P as opposite vertices√ and are otherwise disjoint from P . If there is a point p ∈ P with degree ≥ 2 n in G, then 6.3 CF-Coloring of Range Spaces 75

Figure 6.2: A point p and the neighbors of p in two opposite quadrants in the graph G(P, B2). The doubly-circled points form an independent set in this graph.

√ there are two opposite quadrants around p that contain together at least n neighbors of p in G(P, B2). See Figure 6.2. Suppose, without loss of generality, that these are the upper-right and the lower-left quadrants. The neighbors of p in each of the quadrants form a monotone decreasing sequence. Choosing√ every other element in each sequence yields an independent√ set in G of size at least n/2. Otherwise, all the points of p have degree < 2 n in G. However, in this case, Lemma 6.4(i) implies that there is √ 2 an independent√ set in G of size Ω( n). By Lemma 6.2(ii), (P, B ) can be CF-colored using O( n) colors.

Remark: Noga Alon, Timothy Chan, JánosPach and Geza Tóth[PT03] have independently noticed that the result of Theorem 6.7 can be slightly improved by a polylogarithmic factor, using more involved graph-theoretic arguments [AKS98, PT03]. Their main observation is that the Delaunay graph G(P, B2) has sparse neighborhoods. Namely, for any point p, the subgraph of G induced by the set Np of the neighbors of p has size O(|Np|). The result of [AKS98] implies that if a graph G has maximum degree δ and has ‘sparse neighborhoods’ then G contains an independent set of size Ω(n log δ ). √ δ Choosing δ = n log n we have: If G contains a point with degree more than δ, then by the above analysis G contains an independent set of size Ω(δ). Otherwise, by the sparse neighborhood property of G we have that G contains an independent set of size Ω(n log δ ) = Ω(δ). A simple modification of the proof of Lemma 6.2(ii) implies that the δ √ n number of layers into which P can be decomposed is O( √ ). By Lemma 6.1, (P, B2) √ log n n √ can be CF-colored using O( log n ) colors. Substantially improving the result of Theorem 6.7 is the main open problem that 76 Conflict-Free Coloring Problems

we pose in this chapter, as we currently only have a trivial lower bound of Ω(log n). Using a somewhat diﬀerent approach, we next give an alternative proof of Theo- rem 6.7, which generalizes to higher dimensions:

d d d Theorem³ 6.8.´ Let B be the set of all axis-parallel boxes in IR . Then kopt(n, B ) = O n1−1/2d−1 .

Note that for d = 2 we obtain the same bound as in Theorem 6.7.

d Proof. Let P be a set of n points in IR , and denote the coordinates by x1, . . . , xd. Let P1 be the ordered sequence of the points of P according to their x1-coordinate. At the i-th stage, for i = 2, . . . , d, let Pi be the longest monotone subsequence of Pi−1, according to their x -coordinates. By the Erd˝os-Szekeres Theorem (see, e.g., [Wes01]) i ³p ´ there exists a monotone subsequence of Pi−1 of length Ω |Pi−1| . ³ ´ 1/2d−1 Thus, Pd is a sequence of Ω n points which is monotone in all coordinates (in each coordinate it can be either increasing or decreasing). It is easy to verify that if we pick every other point³ in´ this sequence, we obtain an independent set in d 1/2d−1 G(P, B ) of size |Pd|/2 = Ω n . We thus conclude, by Lemma 6.2(ii), that ³ ´ d 1−1/2d−1 kopt(n, B ) = O n .

Remark: It is easy to construct the CF-coloring provided by Theorem 6.8 in time O(n2−1/2d−1 log n): There are O(n1−1/2d−1 ) iterations, in each of which we compute (d − 1) times a longest monotone subsequence, which can be done in O(n log n) time. In contrast to the rather weak bounds of Theorem 6.7 and Theorem 6.8, we next show that the special case where P is a grid admits a CF-coloring of (optimal) logarithmic size. © ¥ ¦ª Deﬁnition 6.9. The grid G(n, d) is the Cartesian product 1,..., n1/d d.

Lemma 6.10. Let I = B1 be the set of intervals on the real line and let cf(i) be the function deﬁned on the positive integers, so that cf(i) = j + 1 if 2j is the largest power of 2 that divides i.

Then the conﬂict-free chromatic number of (G(n, 1), I) is 1 + blog2 nc, it is realized by cf(·), and this bound is tight. Furthermore, for an interval I = [i, j], the color that appears exactly once in I∩G(n, 1) is the largest number in the set {cf(i), cf(i + 1),..., cf(j)}.

All claims are easy to verify. The lower bound is argued as in the proof of Theo- rem 6.5.

Lemma 6.11. The conﬂict-free chromatic number of (G(n, d), Bd) is at most 1+blog nc.

Pd Proof. For i = (i1, . . . , id) ∈ G(n, d), we deﬁne its color to be f(i) = j=1 cf(ij)−(d−1). Let R be any axis-parallel box, and let Nj be the set of integers in the projection of R onto the j-th axis. Note that, for j = 1, . . . , d, cf(Nj) has a unique maximum in 0 0 this range, by Lemma 6.10. Let ij be the index that realizes it. Clearly, f(i ) is the 6.4 CF-Coloring of Regions 77

0 0 0 0 maximum value achieved by f(·) on R, where i = (i1, i2, . . . , id). Furthermore, no other point of R ∩ G(n, d) realizes this value. Thus f(·) provides the required conﬂict-free coloring.¥ To complete¦ the proof, note that the value of f(·) is bounded from above by d(1 + log(bn1/dc) ) − (d − 1) ≤ 1 + blog nc. Remark: Lemma 6.11 is tight for d = 1 and d = blog nc. For other values of d, one can show a lower bound of blog nc − d. To see this, consider any CF-coloring of G(n, d), and let p be the point with a unique color in the whole grid. Then there is a box that avoids p and contains almost half of the points of G(n, d). Analyzing carefully the number of points remaining in this box, and using induction, we obtain the asserted lower bound.

6.4 CF-Coloring of Regions

In this section we turn to consider the “dual” problem of CF-coloring of regions with respect to points, rather than coloring points with respect to regions, and present one of the main results of this chapter. We introduce a general approach that yields near- optimal bounds on the CF-chromatic number of any finite collection of regions with “low” (usually near-linear) union complexity. Our approach can also be applied to a general geometric range space (not necessarily shrinkable) whose Delaunay graph has “low” complexity. A collection of regions R in IRd induces the following natural abstract range space: The element set of the range space is R and the ranges are all subsets R0 ⊂ R such that there is a point p ∈ IRd for which {r ∈ R|p ∈ r} = R0. In contrast with the range spaces induced by points with respect to discs, the range space induced in this manner by a finite collection of discs is not necessarily shrinkable. We refer the reader to Figure 6.3 for an example of such a collection of discs.1 Note that the general framework suggested by Algorithm 1 cannot be applied to such a collection. Indeed, the three white discs d1, d2, d3 form an independent set in the corresponding Delaunay graph. However, coloring all of them with the same color would give rise to a non-valid coloring (regardless of the colors assigned to the other discs in R), because there exists a point p that belongs to all three of them but does not belong to any other disc in R, so p cannot be served by such a coloring. Hence, a different approach is needed to handle CF-coloring of regions, as will be presented below. Before presenting this main result, we study the problem of CF-coloring of discs in the plane.

6.4.1 From Discs to Halfspaces In what follows, we show that the problem of CF-coloring of n arbitrary discs in the plane reduces to CF-coloring of a set of points P in IR3 with respect to the set of all positive halfspaces. We use a fairly standard dual transformation that maps a point p = (a, b) in IR2 to a plane p∗ in IR3, with the equation z = −2ax − 2by + a2 + b2, and maps a disc

1We are indebted to Shai Zaban for suggesting this example. 78 Conﬂict-Free Coloring Problems

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Figure 6.3: An example of a set R of six discs which form a non-shrinkable range space. The three (white) discs d1, d2, d3 form an independent set in the Delaunay graph induced by R but are not allowed to be colored with the same color.

S in IR2, with center (x, y) and radius r ≥ 0, to a point S∗ in IR3, with coordinates (x, y, r2 − x2 − y2). See [Ede87]. It is easily seen that in this transformation, a point p ∈ IR2 lies inside (resp., on the boundary of, outside) a disc S, if and only if the point S∗ ∈ IR3 lies above (resp., on, below) the plane p∗. Given a collection S = {S1,...,Sn} of n distinct discs in the plane, we map them, ∗ ∗ ∗ 3 using this transform, to obtain a collection S = {S1 ,...,Sn} of n points in IR , such that any CF-coloring of S∗ with respect to all positive halfspaces that uses k colors, induces a CF-coloring of the discs of S with the same set of k colors. Hence, as implied by the result of Theorem 6.6, we have:

Corollary 6.12. Let S = {S1,...,Sn} be a set of n discs in the plane. Then there exists an O(n log n)-algorithm that CF-colors S with O(log n) colors.

6.4.2 CF-Coloring of Regions with Low Union Complexity We next show a strong relation between CF-coloring of regions and the complexity of the union of the underlying regions. As a matter of fact, we present an alternative and more powerful technique to show that any collection of n regions (not necessarily convex) in the plane with the property that any m of them has O(m) union complexity, admits a CF-coloring with O(log n) colors. In particular, since this property holds for discs [KLPS86], it provides an alternative proof of Corollary 6.12. Let R be a family of regions in the plane, such that the complexity of the union of any n regions of R is at most U(n). In the following, we assume that U(n) is a near- 6.4 CF-Coloring of Regions 79

linear function. This holds for pseudo-discs [KLPS86], convex α-fat shapes [ES00], and (α, β)-covered objects [Efr99]. See below (and Chapter 2) for more precise statement

of those bounds.

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Figure 6.4: A set S of discs and an admissible subset Sb (depicted shaded).

Deﬁnition 6.13. For a set S of n regions of R, a subset Sb ⊆ S is admissible (with respect to S), if any p ∈ IR2 satisﬁes one of the following three conditions: 1. p∈ / ∪Sb. 2. p ∈ ∪Sb, but there is only one region of Sb that covers p. 3. p ∈ ∪Sb, and there exists r ∈ S \ Sb, such that p ∈ r. See Figure 6.4. Assume that we are given an algorithm A that computes, for any set of regions S, a non-empty admissible set A(S). We can now use the algorithm A to CF-color the given regions: (i) Compute an admissible set Sb = A(S), and assign to all the regions in Sb the color 1. (ii) Color the remaining regions in S\ Sb recursively, where in the i’th stage we assign the color i to the regions in the admissible set. We denote the resulting coloring by CA(S).

Lemma 6.14. Given a set of regions S, the coloring CA(S) is a valid conﬂict-free coloring of S. Proof. The proof is similar to that of Lemma 6.1. Lemma 6.15. Let R be a set of n regions in the plane and let U(m) denote the maximum complexity of the union of any m regions of R. Let A(R) denote the arrangement of the boundary curves of the regions in R. Then the number F≤k(R) of faces of the arrangement A(R) that are contained inside at most k regions of R is O (k2U(n/k) + n). 80 Conﬂict-Free Coloring Problems

Proof. We may assume that the regions of R are in general position, in the sense that no three distinct boundaries pass through a common point. This can be enforced by a slight perturbation of the curves, which does not decrease F≤k(R). Let S≤k(R) be the set of vertices of the arrangement A(R) that lie in the interior of at most k regions of R. By the probabilistic analysis of Clarkson and Shor [CS89] (see also 2 Theorem 2.15), we have |S≤k(R)| = O (k U(n/k)). We charge a face f contained in at most k regions to its lowest vertex, if ∂f has vertices. Thus, the only faces unaccountable for by this charging scheme are the faces that have no vertices on their boundary. However, it is easy to check that the number of such faces is only O(n), as we can charge such a face to the region of R that forms its outer boundary. Thus 2 F≤k(R) = O(S≤k(R) + n) = O(k U(n/k)) + n).

In what follows, we assume that U(m) ≥ m for any m, so the bound in Lemma 6.15 is O (k2U(n/k)).

Lemma 6.16. Let R be a set of n regions in the plane, so that the boundaries of any pair of them intersect in a constant number of points, and let U(m) denote the maximum complexity of the union of any m regions of ³R. Then´ there exists an admissible set b b n2 S ⊆ R with respect to R, such that |S| = Ω U(n) . Furthermore, such a set can be computed in (randomized expected) O(n3) time, in an appropriate model of computation.

Proof. Let A = A(R) be the arrangement of the regions of R. Place an arbitrary point inside each face of the arrangement A and let P denote the resulting point set. Let χ be a random coloring of the regions of R, by two colors, black and white, where each region is colored independently by choosing black or white with equal probabilities. A point p ∈ P is said to be unsafe if all the regions of R that contain p are colored black. Let PU be the set of unsafe points of P . Let RB be the set of all regions of R which are colored black by χ. We construct a graph G over RB, connecting two regions 0 r, r ∈ RB by an edge if there is an unsafe point p ∈ PU that is contained inside both r and r0. Let e(G) and v(G) denote, respectively, the number of edges and of vertices in G. We claim that, with constant probability, v(G) ≥ n/3 and e(G) = O(U(n)).

Clearly, the condition |RB| = v(G) ≥ n/3 holds with high probability (which tends to 1 when n increases) by the Chernoﬀ inequality (see [AS92]). As for the second claim, for a point p ∈ P , let d(p) denote the number of regions of R that¡ contain¢ it. d(p) d(p) Clearly, the probability that p is unsafe is 1/2 . If p is unsafe, there are 2 pairs ¡d(p)¢ of regions of RB whose intersections contain p, so p induces 2 edges in G. Let Xp ¡d(p)¢ be the random variable having value 0 if p is safe, and 2 if p is unsafe. Clearly, 6.4 CF-Coloring of Regions 81

P e(G) ≤ p∈P Xp. Thus, using linearity of expectation and Lemma 6.15, we have   ¡ ¢ Ã ! X X d(p) Xn X 2  Xn 2 2  i  2 i E[e(G)] ≤ E[Xp] = = O = O i U(n/i) · 2d(p)  2i  2i p∈P p∈P i=2 p∈P i=2 d(p)>1 d(p)=i Ã ! Xn i4 = O U(n) = O (U(n)) . 2i i=2 Thus, by the Markov inequality, it follows that there is a constant c, such that h i P r e(G) ≥ c ·U(n) ≤ 1/4.

It follows that, with constant probability, G has at least n/3 vertices, and its average degree is at most 6c ·U(n)/n. Thus, by Lemma 6.4(i), G contains an independent set of size Ω(n2/U(n)). Let R0 be this independent set. It is easy to verify that R0 is admissible with respect to R. Indeed let f be a face of A(R) that is contained in at 0 least two regions r1, r2 ∈ R , and let p be its representing point. Then p must be safe, so p, and thus f is contained also in a white region, which clearly does not belong to R0. Note that the proof is constructive. Assume a model of computation as in [SA95] in which computing the intersection points of any pair of regions in R, and a few similar operations, can be performed in O(1) time. Then A(R) and P can be constructed in O(n2) randomized expected time [SA95]. We draw the random coloring χ and construct G in O(n3) time (with some care, this can be improved to near quadratic time). In an expected constant number of trials, we obtain G with v(G) ≥ n/3 and e(G) = O(U(n)). Then R0 can be found in O(U(n)) time, as in the preceding sections. This completes the proof of the lemma. Remark: In most applications of the lemma, the running time O(n3) can be improved to near-quadratic, using a more careful implementation of the bottleneck step that identifies all safe points of P . However, we make no attempt in the thesis to work out the details of such improvements. We now present several applications of Lemma 6.16. For the sake of convenience, we repeat some definitions from Section 2.3.1. Definition 6.17 ([KLPS86]). A family R of Jordan regions in the plane is called a family of pseudo-discs if the boundaries of each pair of them intersect at most twice. Theorem 6.18. Let R be a family of n pseudo-discs. Then R admits a CF-coloring with O(log n) colors. Such a coloring can be constructed in randomized expected O(n3) time. Proof. The complexity of the union of any m regions of R is O(m) (see [KLPS86]). Plugging this fact into Lemma 6.16, we have that R contains an admissible set Sb with respect to R of size Ω(n). Applying Lemma 6.14, and arguing as in the proof of Lemma 6.2, we have that R admits a CF-coloring with O(log n) colors. 82 Conflict-Free Coloring Problems

Definition 6.19 ([Efr99]). A planar object c is (α, β)-covered if the following holds: (i) c is simply connected, and (ii) for any point p ∈ ∂c we can place a triangle ∆ fully inside c, such that p is a vertex of ∆, each angle of ∆ is at least α, and the length of each edge of ∆ is at least β times the diameter of c. Theorem 6.20. Let C be a collection of n (α, β)-covered regions in the plane, of finite description complexity, such that the boundaries of each pair of regions of C intersect 3 in at most s points. Then C has a conflict-free coloring using O(βs+2(n) log n log log n) colors, where βs+2(n) = λs+2(n)/n. This coloring can be computed in randomized expected O(n3) time, in an appropriate model of computation.

2 Proof. In this case, U(n) = O(λs+2(n) log (n) log log n) by the result of [Efr99] (see also Theorem 2.19). Thus, by Lemma 6.16, C has an admissible set of size µ ¶ µ ¶ µ ¶ n2 n2 n Ω = Ω 2 = Ω 2 . U(n) λs+2(n) log n log log n βs+2(n) log n log log n Applying the algorithm described in Lemma 6.14, and arguing as in Lemma 6.2, it follows that we have a conﬂict-free coloring of C using

3 O(βs+2(n) log n log log n)

colors. Remark: An earlier result of Efrat and Sharir [ES00] shows that the complexity of the union of n convex α-fat regions2, so that the boundaries of any pair of them intersect at most s times, is O(n1+²), for any ² > 0, where the constant of proportionality depends on ², α, and s. The preceding machinery implies that a collection of n such regions can be CF-colored with O(n²) colors, for any ² > 0. (This result is subsumed, however, by Theorem 6.20.)

6.5 CF-Coloring of Simple Geometric Regions in the Plane

In this section we study some special cases of CF-coloring of regions in the plane, where the underlying regions need not necessarily have low union complexity.

6.5.1 Conﬂict-Free Coloring of Axis-Parallel Rectangles Lemma 6.21. Let R be a set of n axis-parallel rectangles, all intersecting the y-axis. Then there is a CF-coloring of R with O(log n) colors, which can be constructed in randomized expected O(n3) time. Proof. It is easy to verify that the complexity of the union of m such rectangles is O(m). Hence, the result follows immediately from Lemma 6.16 and Lemma 6.14.

2See Section 2.3.1 for th exact deﬁnition of α-fat regions. 6.5 CF-Coloring of Simple Geometric Regions in the Plane 83

Theorem 6.22. Let R be a set of n axis-parallel rectangles in the plane. Then there is a CF-coloring of R using O(log2 n) colors.

Proof. Let ` be a vertical line, such that at most n/2 rectangles of R lie fully to the left of `, and at most n/2 rectangles of R lie fully to its right. Let R0, R1, R2 denote respectively the sets of rectangles crossed by `, lying fully to its left, and lying fully to its right. By Lemma 6.21, we can CF-color the set R0 with O(log n) colors. We color recursively R1 and R2, using the same set of colors in both subproblems, but keeping this set disjoint from the set used to color R0. This gives rise to a coloring of R with a total of O(log2 n) colors, which is easily seen to be a CF-coloring. Again, the proof is constructive, and leads to an O(n3)-randomized expected time algorithm for constructing the coloring.

6.5.2 CF-Coloring of “Well Behaved” Unbounded Regions In this subsection, we study the problem of CF-coloring regions with “well behaved” boundaries. A curve γ is called an x-monotone curve, if it is the graph of a continuous totally deﬁned function. For an x-monotone curve γ in the plane, let γ+ (resp., γ−) denote the set of all points in the plane that lie above (resp., below) γ. Thus for a straight line l, l+ (resp., l−) is the positive (resp., negative) half-plane bounded by l. Let R = {r1, . . . , rn} be a family of n regions in the plane such that each region ri + − is bounded by some x-monotone curve γi (i.e., either ri = γi or ri = γi ). Assume further that the corresponding set of curves Γ = {γ1, . . . , γn} has the property that each pair of curves intersect at most s times, where s is some ﬁxed constant. We refer to such a family of regions as s-order half-planes. Thus, any family of half- planes is a family of 1-order half-planes. Put βs(n) = λs(n)/n, where λs(n) denotes the maximum length of a Davenport- Schinzel sequence of order s with n symbols.

Theorem 6.23. Let R = {r1, . . . , rn} be a family of n s-order half-planes. There exists a CF-coloring of R with O(βs(n) log n) colors and such a coloring can be found in (randomized expected) O(nβs+2(n) log n) time. Proof. Since we may color separately the set of positive halfplanes and the set of negative halfplanes, using diﬀerent colors, we may assume without loss of generality that + each region ri ∈ R is a positive halfplane ri = γi . Let Γ = {γ1, . . . , γn} denote the set of the boundary curves. Note that for a given point p, the set of regions in R that contain p is equal to the set of regions whose boundary curves are intersected by a vertical downward-directed ray emanating from p. Let Γp denote the set of these curves. Let E(Γ) denote the set of all distinct pairs {γi, γj} ∈ Γ×Γ for which there is a point p such that Γp = {γi, γj}. Any such point p lies in a face belonging to the second belt of A(Γ), consisting of those faces with exactly two curves below them. Hence the size of E(Γ) is at most the number of such faces. It is easily seen, using the Clarkson-Shor technique [CS89], that the number of these faces has the same asymptotic upper bound as that for 84 Conﬂict-Free Coloring Problems

the complexity of the lower envelope of the curves of Γ, that is, |E(Γ)| = O(λs(n)) (see [SA95]). Let G(Γ) = (Γ,E(Γ)) be the graph on Γ whose edges are the pairs in E(Γ). (This is in fact the Delaunay graph of R.) Since |E(Γ)| = O(λs(n)), it follows, using 2 Lemma 6.4(i), that G(Γ) contains an independent set of size Ω(n /λs(n)) = Ω(n/βs(n)). Let Γ1 ⊂ Γ be an independent set of size Ω(n/βs(n)). It is easily seen that the the set of regions corresponding to Γ1 is admissible with respect to R. Thus, using the algorithm of Lemma 6.14 it is easily seen that R can be CF-colored with O(βs(n) log n) colors. The algorithmic construction of such a coloring can be done using the method of Agarwal et al. [AdBMS98] for constructing the ﬁrst two levels of A(Γ) in O(nβs+2(n) log n).

The above approach also works for any ﬁnite family R of star-shaped regions, with respect to a common point o, for which the union of any subset has linear (or near linear) complexity. The simplest way of showing this is to use polar coordinates (ρ, θ) about o, and interpret each region ri ∈ R as the portion of the polar plane lying below the function ρ = γj(θ) that represents ∂rj in polar coordinates. The preceding analysis then applies verbatim to this case. Remark: Recall that, in general, the range space induced by a collection R of regions is not necessarily shrinkable as shown in Figure 6.3. However, it is shrinkable in the special case at hand as can be easily veriﬁed. In analogy with the lower bound of Theorem 6.5, which applies to CF-coloring of range spaces, we next provide a similar lower bound for CF-coloring of regions.

Theorem 6.24. There exists a collection H of n half-planes, for which Ω(log n) colors are needed in any CF-coloring of H.

Proof. We use the standard dual transformation, mentioned in Section 3.2 that maps a line l to a point l∗ and a point p to a line p∗, such that p lies above (resp., below) l if ∗ ∗ and only if the line p lies© above (resp.,ª below) the point l . It is easily verified that any + + CF-coloring of a set H = l1 , . . . , ln of n positive half-planes is equivalent to that of a ∗ ∗ CF-coloring of a range space (P, R), where P = {l1, . . . , ln} is the set of dual points of the boundary lines of the half-planes in H, and R is the set of all negative half-planes. Thus, it suffices to show that for any integer n, there exists a set P of n points in the plane such that any CF-coloring of P with respect to negative half-planes needs at least Ω(log n) colors. Such a construction can be obtained by an obvious modification of the one provided in the proof of Theorem 6.6.

Similar constructions show that there exists a collection R of n axis-parallel rectangles for which Ω(log n) colors are needed in any CF-coloring of R. This still leaves a logarithmic gap with the upper bound of Theorem 6.22. In the context of range spaces, similar constructions of a set of n points in the plane show that in any CF-coloring of the given points, Ω(log n) colors are needed when the ranges are axis-parallel rectangles. 6.6 Miscellaneous CF-Coloring Problems 85

6.6 Miscellaneous CF-Coloring Problems

6.6.1 CF-Coloring of Points with respect to Balls and Halfs- paces in Higher Dimensions

Lemma 6.25. Let R be the set of balls in three dimensions. Then kopt(n, R) = n. The same holds for the set R of halfspaces in IRd, for d > 3.

Proof. Take P to be a set of n points on the positive portion of the moment curve γ = {(t, t2, t3)|t ≥ 0} in IR3. It is easy to verify that any pair of points p, q ∈ P are connected in the Delaunay triangulation of P [Eri01], implying that there exists a ball whose intersection with P is {p, q}. Thus, all points must be colored using different colors. The second claim follows by lifting P into the standard paraboloid in IR4 by the map (x, y, z) 7→ (x, y, z, x2 + y2 + z2). A ball in IR3 is mapped to a halfspace in IR4 so that a point p lies in the ball if and only if its image lies in the halfspace. It follows that n colors are necessary in any CF-coloring of the image of P . This clearly extends to any dimension d ≥ 4. Remark: The conflict-free chromatic number of the range space of points and fat triangles in the plane is n. Indeed, let P be a set of n points placed along a circle. It is easy to verify that for every pair of points p, q ∈ P , one can construct a fat triangle that contains p and q, but does not contain any other point of P . Hence, as above, n colors are needed. In Section 6.7 we relax the notion of CF-coloring, so that a range r is “served” if there exists a color that appears at most k times in the given range for some fixed k ≥ 2 (rather than exactly once). We show that the problem of coloring the above range spaces becomes non-trivial and interesting in this setting. An analogous relaxation of CF-coloring of regions is also studied.

6.6.2 CF-Coloring of Half-Slabs Deﬁnition 6.26. A half-slab is an axis-parallel rectangle [a, b] × [c, d] where at least one of the values {a, b, c, d} is ±∞.

In contrast with Theorem 6.7, we have:

Lemma 6.27. Let R be the set of all axis-parallel half slabs in the plane, and P a set of n points. The CF-chromatic number of (P, R) is Θ(log n).

Proof. We construct the Delaunay graph G = G(P, R) as before, consisting of all pairs of points (p, q) for which there is a half-slab sp,q that contains only p and q on its boundary, and its interior is empty. Consider the subgraph of G induced by pairs p, q for which sp,q is unbounded in the positive x-axis. See Figure 6.5 for an illustration. We 0 slide sp,q in the negative x-direction until it touches a third point p ∈ P . We assume that such a point exists. (Otherwise, the number of pairs p, q ∈ P for which there is an

86 Conﬂict-Free Coloring Problems

¢¡

Figure 6.5: A half-slab parallel to the x-axis.

infinite slab enclosed by two horizontal lines, that contains only p and q on its boundary 0 is n−1.) We charge sp,q to the point p . It is easily verified that this charging is unique: Take the ray emanating from p0 in the positive x-direction and slide it up and down in the y-direction until it hits a point of P ; these two points are p and q. Applying the same charging to slabs which are unbounded in the negative x-direction or the negative or positive y-direction, we conclude that the number of edges in G is O(n). Applying Lemma 6.4(i) to G, we conclude that G contains an independent set of size Ω(n), and by Lemma 6.2, (P, R) has a conflict-free coloring with O(log n) colors.

6.7 Relaxing the Notion of Conﬂict-Free Coloring

In this section we generalize the notion of CF-coloring of a range space and show a relation between the problem of CF-coloring a range space and its VC-dimension. We also generalize the notion of CF-coloring of regions. To simplify the presentation, we ignore in this section the issue of algorithmic construction of the coloring. Neverthe- less, all upper bounds in this section are constructive, and can be easily computed in polynomial time.

6.7.1 k-CF-Coloring of a Range-space Deﬁnition 6.28. k-CF-coloring of a range space: Let (P, R) be a range space in IRd. A function χ : P → {1, . . . , i} is a k-CF-coloring of (P, R) if for every r ∈ R with r ∩ P 6= ∅ there exists a color j such that 1 ≤ |{p ∈ P ∩ r|χ(p) = j}| ≤ k; that is, for every possible nonempty range r there exists at least one color j such that j appears (at least once and) at most k times among the colors assigned to points of P ∩ r.

Let kopt(n, k, R) denote the minimum number of colors needed for a k-CF-coloring of (P, R), maximized over all sets P of size n. Note that a 1-CF-coloring of a range space is just a CF-coloring. As mentioned in Lemma 6.25, there exist range spaces (P, R) consisting of n points in IR3, where the ranges are induced by balls, for which any CF-coloring (i.e., 1-CF- coloring) of (P, R) needs n colors. However, the bound drastically improves if we consider k-CF-coloring with k ≥ 2. 6.7 Relaxing the Notion of Conﬂict-Free Coloring 87

3 1/k Theorem 6.29. Let R be the set of all balls in IR . Then kopt(n, k, R) = O(n ), for any ﬁxed constant k ≥ 1. Proof. The proof technique is a generalization of the ideas developed in Section 6.2. We construct a (k + 1)-uniform hypergraph H = (P, E), where E is the collection of all subsets of P of size k + 1 that are realizable by a range in R. By the Clarkson-Shor technique, it is easy to see that |E| = O(n2), where the constant of proportionality depends on k. Thus, the average degree of H is O(n) and therefore, by Lemma 6.4(ii), there exists an independent set P 0 ⊂ P of size Ω(n1−1/k). (Note that independence means that any ball that contains at least k + 1 points of P 0, must also contain a point from P \ P 0; this equivalence follows by an appropriate extension of the monotonicity property of balls.) We can color all points of P 0 by a single color, say 1, and iterate on P \ P 0, similar to Algorithm 1. Thus, the total number of colors we use is O(n1/k). It is easy to see (similar to Lemma 6.1) that this coloring is a valid k-CF-coloring of (P, R).

6.7.2 k-CF-Coloring of Range Spaces with Finite VC-Dimension Definition 6.30. Let S = (X, R) be a range space. The Vapnik-Chervonenkis dimension (or VC-dimension) of S, denoted by VC(S), is the maximal cardinality of a subset P 0 ⊂ P such that {P 0 ∩ r|r ∈ R} = 2P 0 (such a subset is said to be shattered). If there are arbitrarily large shattered subsets in X then VC(S) is defined to be ∞. See, e.g., [AS92, PA95] for additional details. There are many range spaces with finite VC-dimension that arise naturally in combinatorial and computational geometry. One such example is the range space d d S = (IR , Hd), where Hd is the family of all (open) halfspaces in IR . Any set of d + 1 affinely independent points is shattered in this space, and, by Radon’s theorem, no set of d + 2 points is shattered. Therefore VC(S) = d + 1. As a matter of fact, all range spaces used in this chapter have finite VC-dimension. Definition 6.31. Let (P, R) be a range space with |P | = n and let 0 < ² ≤ 1. A subset P 0 ⊂ P is called an ²-net for (P, R) if for every range r ∈ R with |r ∩ P | ≥ ²n we have r ∩ P 0 6= ∅. An important consequence of the finiteness of the VC-dimension is the existence of small ²-nets, as shown by Haussler and Welzl [HW87], where the notion of VC-dimension of a range space was introduced to computational geometry. Theorem 6.32. For any range space S = (P, R) with finite VC-dimension d and for 0 d d any ² > 0, there exists an ²-net P ⊂ P of size O( ² log ² ). Since all the range spaces studied in this chapter have finite VC-dimension, and since some of them can be CF-colored only with n colors, there is no direct relationship between a finite VC-dimension of a range space and the existence of a CF-coloring of that range space with a small number of colors. In this subsection we show that such a relationship does exist, if we consider k-CF-coloring with a reasonably large k. 88 Conflict-Free Coloring Problems

We ﬁrst introduce a general framework for k-CF-coloring of a range space S = (X, R). Deﬁnition 6.33. A subset X0 ⊂ X is k-admissible with respect to S if for any range r ∈ R with |r ∩ X0| > k we have r ∩ (X \ X0) 6= ∅. Note that, assuming an appropriate monotonicity property of the ranges in R, a k-admissible set is simply an independent set in the hypergraph (X, E), where E is the set of all hyperedges consisting of k + 1 elements of X that can be realized by a range in R. Assume that we are given an algorithm A that computes, for any range space S = (X, R), a non-empty k-admissible set X0 = A(S). We can now use the algorithm A to k-CF-color the given range space: (i) Compute an admissible set X0 = A(S), and assign to all the elements in X0 the color 1. (ii) Color the remaining elements in X \ X0 recursively, where in the i’th stage we assign the color i to the points in the resulting k-admissible set. We denote the resulting coloring by CA(S).

Lemma 6.34. Given a range space S = (X, R), the coloring CA(S) is a valid k- conﬂict-free coloring of S. The proof is similar to that of Lemma 6.1. Lemma 6.35. Let S = (X, R) with |X| = n be a ﬁnite range space with VC-dimension d. Then there exists a constant c = c(d) such that for k ≥ 2c log n there exists a k-admissible set X0 ⊂ X with respect to S of size at least n/2.

k Proof. Suppose that k ≥ 2c log n and put ² = n . By Theorem 6.32 there exists a c 1 cn n constant c = c(d) and an ²-net A ⊂ X for S of size at most ² log ² = k log k . We claim that X0 = X \ A is k-admissible with respect to S. Indeed, if a range r ∈ R contains more than k = ²n elements of X0 then, by deﬁnition, r must contain an element of 0 0 cn n cn A = X \ X . Observe that |X | ≥ n − k log k > n − k log n ≥ n/2 for k ≥ 2c log n. This completes the proof of the lemma. Theorem 6.36. Let S = (X, R) with |X| = n be a ﬁnite range space with VC- dimension d. Then there exists a constant c = c(d) such that for k ≥ 2c log n there exists a k-CF-coloring of S with O(log n) colors. Proof. By Lemma 6.35 S contains a k-admissible set of size at least n/2. Plugging this fact to the algorithm suggested by Lemma 6.34 completes the proof of the theorem. As remarked above, Theorem 6.36 applies to all the range spaces studied in this chapter.

6.7.3 k-CF-Coloring of Regions Deﬁnition 6.37. k-CF-coloring of regions: Let R be a collection of regions in IRd. A function χ : R → {1, . . . , i} is a k-CF-coloring of R if for every point p ∈ ∪R there exists a color j such that 1 ≤ |{r ∈ R|p ∈ r, χ(r) = j}| ≤ k; that is, for every possible 6.7 Relaxing the Notion of Conﬂict-Free Coloring 89

point p in the union of R there exists at least one color j such that j appears (at least once and) at most k times among the colors assigned to the regions of R that contain p. As above, we note that a 1-CF-coloring of a set of regions R is just a CF-coloring of R. Consider a CF-coloring of a set of balls in IR3. Note that the union of a set of n balls can have Ω(n2) complexity and one cannot apply the technique developed in Section 6.4 to obtain non-trivial bounds on the number of colors needed for a 1-CF-coloring of such a set of balls, or other regions with high union complexity. However, as we will show in this section, one can obtain non-trivial bounds on the number of colors needed for k- CF-coloring a set of regions in IR3 with near-quadratic union complexity, for any k ≥ 2. The approach that we use generalizes to any ﬁxed dimension. Let R be a family of regions in IR3, such that the complexity of union of any n regions of R is at most U(n). In the following, we assume that U(n) is a monotone increasing function of n and that U(n) = Ω(n2). This holds for balls with U(n) = Θ(n2) (see, e.g., [SA95]).

Deﬁnition 6.38. For a set S of n regions, a subset Sb ⊆ S is k-admissible with respect to S, if any p ∈ IR3 satisﬁes one of the following three conditions: 1. p∈ / ∪Sb.

2. p ∈ ∪Sb, but there are at most k regions of Sb that cover p.

3. p ∈ ∪Sb, and there exists r ∈ S \ Sb, such that p ∈ r. Assume that we are given an algorithm A that computes for any set S of regions in a given family a non empty k-admissible set A(S) with respect to S. We can then use the algorithm A for k-CF-coloring the given regions as follows: (i) Compute a k- admissible set Sb = A(S) with respect to S, and assign to all the regions in Sb the color 1. (ii) Color the remaining regions in S\ Sb recursively, using colors ≥ 2. We denote the resulting coloring by CA(S).

Lemma 6.39. Given a set of regions S, the coloring CA(S) is a valid k-CF-coloring of S. The proof is similar to that of Lemma 6.1 and Lemma 6.14. The following result extends Lemma 6.15 to three dimensions. Lemma 6.40. Let R be a set of n regions in IR3 of constant description complexity and let U(m) denote the maximum complexity of the boundary of the union of any m 2 regions of R, with U(m) = Ω(m ). Then the number F≤i(R) of 3-dimensional cells of the arrangement A(R) that are contained in at most i regions of R is O (i3U(n/i)).

Proof. Let S≤i(R) be the set of vertices of the arrangement A(R) (of the boundary surfaces of the regions in R) that lie in the interior of at most i regions of R. By the 3 Clarkson-Shor technique [CS89], we have |S≤i(R)| = O (i U(n/i)). We charge a cell 90 Conﬂict-Free Coloring Problems

contained in at most i regions to its lowest vertex, assuming it has a vertex. Thus, the only cells unaccountable for by this charging scheme are the cells that have no vertices on their boundary. However, it is easy to check that the number of such cells is bounded by O(n2). Thus

2 3 2 3 F≤i(R) = O(S≤i(R) + n ) = O(i U(n/i) + n ) = O(i U(n/i)), by our assumptions on U(n). Lemma 6.41. Let R be a set of n regions in IR3, and let U(m) denote the maximum complexity of the union of any m regions of R, such that U(m) = Ω(m2) and U(·) is b monotone increasing.³ Then´ there exists a k-admissible set S ⊆ R with respect to R, such that |S|b = Ω n1+1/k . U(n)1/k Proof. The proof follows closely the ideas of the proof of Lemma 6.16 with a slight twist. Let A = A(R) be the arrangement of the (boundary surfaces of the) regions of R. Place an arbitrary point inside each (three-dimensional) cell of the arrangement A and let P denote the resulting point set. Let χ be a random coloring of the regions of R, by two colors, black and white, where each region is colored independently by choosing black or white with equal probabilities. A point p ∈ P is said to be unsafe if all the regions of R that contain p are colored black. Let PU be the set of unsafe points of P . Let RB be the set of all regions of R which are colored black by χ. We construct a (k + 1)-uniform hypergraph H over RB, whose set of hyperedges consist of all (k + 1)-tuples of regions r1, . . . , rk+1 ∈ RB for k+1 which there is an unsafe point p ∈ PU in ∩j=1 rj. Let e(H) and v(H) denote respectively, the number of hyperedges and of vertices of H. We claim that, with constant probability, v(H) ≥ n/3 and e(H) = O(U(n)). Clearly, the condition |RB| = v(H) ≥ n/3 holds with high probability by the Cher- noﬀ inequality (see, e.g., [AS92]). Similar to the proof of Lemma 6.16, the probability that p is unsafe is 1/2d(p), where d(p) is the number of regions containing p. If p is ¡d(p)¢ unsafe, there are k+1 (k + 1)-tuples of regions of RB whose intersection contains p, ¡d(p)¢ so p induces k+1 hyperedges in H. Let Xp be the random variable having value 0 if ¡d(p)¢ P p is safe, and k+1 if p is unsafe. Clearly, e(H) ≤ p∈P Xp. Thus, using linearity of expectation and Lemma 6.40, we have   ¡ ¢ X X d(p)  Xn X k+1  k+1  i  E[e(H)] ≤ E[Xp] = = O = 2d(p)  2i  p∈P p∈P i=k+1 p∈P d(p)≥k+1 d(p)=i Ã ! Ã ! Xn ik+1 Xn ik+4 O i3U(n/i) · = O U(n) = O (U(n)) . 2i 2i i=k+1 i=k+1 Thus, by the Markov inequality, it follows that there is a constant c, such that h i P r e(H) ≥ c ·U(n) ≤ 1/4. 6.7 Relaxing the Notion of Conﬂict-Free Coloring 91

It follows that, with constant probability, H has at least n/3 vertices, and its average degree is at most (k+1)3c·U(n)/n. Thus, by Lemma 6.4(ii), H contains an independent µ ¶ ³ ´ set of size Ω n = Ω n1+1/k . It is easy to verify that any such independent U(n) 1/k U(n)1/k ( n ) set is k-admissible with respect to R. This completes the proof of the lemma. Note that when U(n) = O(n2) we have a k-admissible set of size Ω(n1−1/k).

Theorem 6.42. Let R be a set of n balls in IR3. For any k ≥ 2, there exists a k-CF- coloring of R with a total of at most O(n1/k) colors.

Proof. By Lemma 6.41 there exists a k-admissible set R0 with respect to R of size Ω(n1−1/k). Plugging this fact into the algorithm suggested by Lemma 6.39 completes the proof. Remark: A closer inspection of the analysis of the proof of Lemma 6.41 shows that the lemma generalizes to any dimension d ≥ 3, provided that we assume that U(m) = Ω(md−1). In this case, we obtain:

Theorem 6.43. Let R be a set of n regions in IRd with the property that the complexity of the union of any m regions of R is at most U(m), where U(m) = Ω(md−1) and is b monotone¯ increasing.¯ ³ Then´ there exists a k-admissible set S ⊆ R with respect to R, ¯ ¯ 1+1/k such that ¯Sb¯ = Ω n . U(n)1/k Remark: The condition that U(m) = Ω(md−1) can be dropped, using a more careful analysis, based on the Clarkson-Shor technique. We omit details of this improvement. The following is an example of an application of Theorem 6.43.

Theorem 6.44. Let R be a set of n regions in IRd with the property that the complexity of the union of any m regions of R is at most O(md−1). Then there exists a k-CF- d−2 coloring of R with a total of O(n k ) colors, for any constant k ≥ d − 1.

Proof. By Theorem 6.43 there exists a k-admissible set R0 with respect to R of size 1− d−2 Ω(n k ). Plugging this fact into the algorithm suggested by Lemma 6.39 completes the proof. 92 Conﬂict-Free Coloring Problems Bibliography

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