Combinatorial Problems in Computational Geometry

Transcription

1 Tel-Aviv University Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science Combinatorial Problems in Computational Geometry Thesis submitted for the degree of Doctor of Philosophy by Shakhar Smorodinsky Under the supervision of Prof. Micha Sharir Submitted to the Senate of Tel-Aviv University June 2003

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3 The work on this thesis was carried out under the supervision of Prof. Micha Sharir iii

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5 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

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7 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 János Pach 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, János Pach, 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, János Pach, 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

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9 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 Applications: Suppose we draw a simple undirected graph G = (V, E) in the plane using points to represent vertices, and Jordan arcs connecting them to represent edges. Assume that G has n vertices and m edges and that m 4n. Then, using a planarity argument, there must exist two crossing arcs in this drawing. This fact can be exploited to show that the number of such crossings is Ω(m 3 /n 2 ), no matter how the graph is drawn. The proof of this Crossing Lemma is due to Leighton [Lei83] and to Ajtai et al. [ACNS82]. A probabilistic proof of this fact was entitled A proof from the book [AZ98]. Adapting and extending the proof technique of the Crossing Lemma, we provide improved asymptotic bounds on well-studied geometric combinatorial problems, such as the k-set problem (Chapter 4), the complexity of polytopes spanned by sets of points in the plane and in space (Chapter 3), etc. In Chapter 2 we provide some sharp asymptotic Ramsey type theorems for intersection patterns of nice objects that are spanned by finite point sets: For example, we prove that for any dimension d, there exists a constant c = c(d) such that for any set P of n points in IR d and any set S of m > cn distinct balls, each bounded by a sphere passing through a distinct pair of points of P, there exists a subset S of S of size at least Ω(m 2 /n 2 ) 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

10 convex polytopes that are spanned by n points in IR 3, where the complexity of a polytope is the total number of its facets. We show an upper bound of O(n 2 k 1/3 ). 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 IR 3. 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 IR 3, i.e., the subsets P of P of cardinality k for which there exists a halfspace H such that P = P H, is O(nk 3/2 ). This improves the previously best known bound of O(nk 5/3 ) (see [DE94, AACS98]). The technique used to obtain this result is to establish a Crossing Lemma for triangles and points in IR 3, 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 Matoušek [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(nk 1/3 ) 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 pseudolines 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

11 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 IR 3 for which in any CF-coloring, n colors are necessary, but one can k-cfcolor any collection of n balls in IR 3 with O(n 1/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

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13 Contents Acknowledgments Abstract vii ix 1 Introduction 1 2 Point-Selection Lemmas Introduction Discs Spanned by Points in IR Pseudo-discs and Points in IR Yet Another Approach for Planar Selection Lemmas Balls and Points in Higher Dimensions Lines Stabbing Discs in IR Upper Bounds Axis-Parallel Rectangles An Upper Bound Open Problems Triangles and Points in IR Introduction Polygons in Point Sets and Concave Chains in Arrangements of Lines Triangles and Points in IR Polytopes Spanned by Point Sets Lower Bound Open Problems k-sets in IR Introduction k-sets and Triangles in IR An Overview of Our Technique Proof of the Theorem Proof of Theorem Open Problems xiii

14 5 Generalized Geometric Graphs and Pseudo-line Arrangements Introduction Drawing Pseudo-line Graphs The Complexity of a k-level in Pseudo-line Arrangements Yet Another Proof for Incidences and Many Faces in Pseudo-line Arrangements Graphs in Pseudo-line Arrangements without Anti-Diamonds Pseudo-line and Thrackles Conflict-Free Coloring Problems Introduction A General Framework CF-Coloring of Range Spaces Coloring Points in the Plane with Respect to Discs Coloring Points in IR 3 with Respect to Halfspaces Axis-parallel Rectangles CF-Coloring of Regions From Discs to Halfspaces CF-Coloring of Regions with Low Union Complexity CF-Coloring of Simple Geometric Regions in the Plane Conflict-Free Coloring of Axis-Parallel Rectangles CF-Coloring of Well Behaved Unbounded Regions Miscellaneous CF-Coloring Problems CF-Coloring of Points with respect to Balls and Halfspaces in Higher Dimensions CF-Coloring of Half-Slabs Relaxing the Notion of Conflict-Free Coloring k-cf-coloring of a Range-space k-cf-coloring of Range Spaces with Finite VC-Dimension k-cf-coloring of Regions Bibliography 92 xiv

15 Chapter 1 Introduction 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. Selection lemmas. In Chapter 2 we study several point selection problems of the following flavor. Let P be a set of n points in IR d, 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 Matoušek [Mat02]. Bárány [Bár82] has shown that for any finite set P of n points in IR d there is always a point that lies in the interior of Ω( ( n d+1) ) = Ω(n d+1 ) 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(n 3 ) 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 IR 2 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(t 3 /(n 6 log 5 n)). Their motivation was to derive an upper bound on the number of halving planes of a finite set of points in IR 3 (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 IR 3 determines at most O(n 8/3 log 5/3 n) halving planes. This result is improved in this thesis to O(n 5/2 ). 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].

16 2 Introduction Their goal was to reduce the size of Delaunay triangulations for finite point sets in IR 3. 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 IR 3, 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 IR d 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(m 2 /(n 2 log 2d 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(m 2 /(n 2 log 2d 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 E E 4 V. Then, in any drawing of G in the plane there must be at least 3 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

17 Introduction 3 classified as a proof from THE BOOK (in Erdős 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 pointselection theorems involving points and balls in IR d, 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 IR 2 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 IR d and any set S of m > cn distinct balls, each bounded by a sphere passing through a distinct pair of points of P, there exists a subset S of S of size at least Ω(m 2 /n 2 ) 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(m 3/2 /n 3/2 ) 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 m 2 subset of R of size at least 512n 2 log 2 with nonempty intersection. (A different 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 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(nk 1/3 ) for k n and Θ(n 2/3 k 2/3 ) for k n. Next, we generalize the Crossing Lemma to three dimensions as follows: We say that two triangles 1, 2 in IR 3, 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 1 The constant 64 has recently been improved; see [PT97].

18 4 Introduction the relative interior of 2. We show that in any set of t > 2n 2 triangles spanned by n points in IR 3 in general position, there must be at least Ω(t 3 /n 4 ) crossing pairs of triangles (with a common vertex). This enables us to prove that for any set of n points in IR 3 and any set of t > 2n 2 triangles spanned by those points, there exists a line that stabs Ω(t 3 /n 6 ) 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 Lovász Lemma for halving triangles, provides an O(n 8/3 ) upper bound on the number of halving triangles in a set of n points in IR 3. 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 IR 3. Specifically, we show that the total complexity k such polytopes is O(n 2 k 1/3 ). 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(n 3/2 k 1/2 ). k-sets. Let S be a set of n points in IR d in general position. A k-set of S is a subset S S such that S = S H for some halfspace H and S = 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(nk 1/3 ) (see [Dey98]) and n 2 Ω( log k) (see [Tót01]), respectively. The best known upper bound in IR 3 is O(nk 5/3 ) [DE94], and the best known lower bound is Ω(nk2 Ω( log k) ) [Tót01]. In Chapter 4 we obtain an improved upper bound of O(nk 3/2 ) on the number of k-sets in a set of n points in IR 3. 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 n 2 triangles spanned by n points in IR 3 that has the antipodality property, then the number of crossing pairs of triangles (with a common vertex) is Ω(t 2 /n), improving significantly the general lower bound Ω(t 3 /n 4 ) mentioned above. This implies that there is a line that stabs Ω(t 2 /n 3 ) of the given triangles. Again, combined with Lovász Lemma for halving triangles, this provides the desired bound t = O(n 5/2 ). 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 Matoušek [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. 2 It is indeed sufficient to establish this bound for the case k = (n 3)/2 of halving triangles.

19 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, γ ) of distinct pseudo-lines in Γ, we denote by W (γ, γ ) the double wedge formed between γ and γ, 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, γ ) 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, γ ) and (δ, δ ) of G form a diamond if the point γ γ is contained in the double wedge W (δ, δ ), and the point δ δ is contained in the double wedge W (γ, γ ). See Figure 5.1(i). Definition 1.3. We say that two edges (γ, γ ) and (δ, δ ) of G form an anti-diamond if the point γ γ is not contained in the double wedge W (δ, δ ), and the point δ δ is not contained in the double wedge W (γ, γ ); that is, γ γ lies in W c (δ, δ ) and δ δ lies in W c (γ, γ ). 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.

20 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(nk 1/3 ) 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 (a, b) and (c, d) y = cx + d, presented in [Ede87], and map each edge (γ, γ ) of G to the straight segment connecting the points dual to γ and γ, 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(nk 1/3 ), 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, e is said to be parallel (or avoiding) if the line containing e does not cross e and the line containing e does not cross e. (For straight edges, this is equivalent to the condition that e and e 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(m 2/3 n 2/3 + m + n).

21 Introduction 7 (b) The maximum number of edges bounding m distinct faces in an arrangement of n pseudo-lines is Θ(m 2/3 n 2/3 + n). Theorem 1.7 was originally obtained by Clarkson et al. [CEG + 90], extending the original result of Szemerédi and 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) 2/3 n 2/3 ). 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.

22 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 that for any n there exists a collection R of n balls in IR 3 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(n 1/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 IR d 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, 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 IR 3 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 IR d can be CF-colored with O(n 1 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-cfcoloring 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]. 3 As a matter of fact, antennas are generally directional, so the region controlled by an antenna is a circular sector rather than a full disc.

23 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 IR d 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

24 10 Point-Selection Lemmas Table 2.1: Summary of point selection bounds. objects/spanned by dim prev. bound new bound stab. pt in P discs/point pairs 2 Ω( m 2 n 2 log 4 n ) m2 Ω( ) n 2 discs/triples of points 2 - Ω( m3/2 pseudo-discs/point pairs 2 - Ω( m2 pseudo-discs/triples of points 2 - Ω( m3/2 balls/point pairs d Ω( m 2 m2 Ω( n 2 log 2d (m 2 /n) ) ) n 3/2 ) n 2 ) n 3/2 ) n 2 lines stabbing discs/point pairs 3 - Ω( m2 ) - n 2 m axis-parallel rectangles 2 Ω( 2 ) O( m 2 ) no n 2 log 2 n n 2 log(n 2 /m) yes yes yes yes 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(m 2 /n 2 ) (resp., Ω(m 3/2 /n 3/2 )) discs. A simple application of the latter bound is an alternative derivation of the bound O(nk 2 ) 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 IR d spanned by distinct pairs of points of P. We show that there exists a point (not necessarily of P ) that lies inside Ω(m 2 /n 2 ) balls. We also study a variant where we have n points in IR 3 and m distinct discs, each spanned by a distinct pair of points. We show that there exists a line that stabs Ω(m 2 /n 2 ) 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(m 2 /n 2 log 2 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(m 2 /(n 2 log(n 2 /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-(m 2 /n 2 ) 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].

25 2.2 Discs Spanned by Points in IR Discs Spanned by Points in IR 2 Theorem 2.1. Let P be a set of n points and let D be a set of m 4n distinct discs in IR 2. (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(m 2 /n 2 ) 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(m 3/2 /n 3/2 ) 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(m 2 /n 2 ) discs in case (i), or O(m 3/2 /n 3/2 ) 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 X configurations, where X X + 1. We have m 1 3n = k, so we can apply the induction hypothesis to obtain X m 1 3n. Thus X X + 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 of the points in P by choosing each point independently with some fixed

26 12 Point-Selection Lemmas probability p (to be determined later on). Let D denote the subset of discs in D, all of whose defining points are in P. Put n = P ; m = D, and let X denote the number of configurations all of whose defining points are in P. Consider first case (i) of the theorem. By Lemma 2.2 we have X m 3n. Note that X, m and n are random variables, so the above inequality holds for their expectations as well. Hence (using linearity of expectation), E[X ] E[m ] 3 E[n ]. It is easily seen that E[n ] = pn. We have E[m ] = p 2 m and E[X ] = p 3 X. Indeed, the probability that a given disc d D belongs to D is the probability that the two points defining d are chosen in P, which is p 2 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 X is p 3. Substituting these values in the above inequality, we get p 3 X p 2 m 3pn, or X m 3n. This inequality holds for any 0 < p 1, and p p 2 we choose p = 4n/m (by assumption, p 1) to obtain X m2. By the pigeonhole 16n principle, one of the points in P is covered by at least X m2 discs. This completes n 16n 2 the proof of case (i) of the theorem. For case (ii), we have X m 2n, E[m ] = p 3 m, and E[X ] = p 4 X, which implies that p 4 X p 3 m 2pn, or X m 2n. This inequality holds for any 0 < p 1, and p p 3 we choose p = 2 n/m (again, p 1), to obtain X m3/2. As above, one of the points 4n 1/2 in P is covered by at least X m3/2 of the discs. This completes the proof of case (ii) n 4n 3/2 of the theorem. The proofs of the worst-case optimality of these bounds are delegated to Section 2.5. Remark: A simple application of the above analysis is an alternative derivation of the bound O(nk 2 ) 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(m 3/2 /n 1/2 ). 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(nk 2 ). (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(nk 2/3 ), 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 IR 2 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 IR 2.