Geometric Permutations and Common Transversals

7 downloads 0 Views 304KB Size Report
Consider geometric permutations induced by line transversals to families A of ... unit discs in the plane is 2, improving a previous upper bound of 3 given in 15]. ..... Proof: Consider two lines l1; l2 that induce the same geometric permutation on a ..... Assume to the contrary that there are two pairs (a; b) and (c; d) that consist of ...
TEL-AVIV UNIVERSITY RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES SCHOOL OF MATHEMATICAL SCIENCES

Geometric Permutations and Common Transversals Thesis submitted in partial ful llment of the requirements for the M.Sc. degree of Tel-Aviv University by Shakhar Smorodinsky

The research work for this thesis has been carried out at Tel-Aviv University under the direction of Prof. Micha Sharir July 1998

Contents 1 Introduction 1.1 1.2 1.3 1.4 1.5

Overview . . . . . . . . . . . k-Transversals . . . . . . . . Hyperplane Transversals . . Spaces of Line Transversals Geometric Permutations . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

2 2 3 4 6 7

2 Geometric Preliminaries

12

3 Geometric Permutations in The Plane

14

2.1 Geometric Permutations and Orientations Space . . . . . . . . . . . . . . . . 12 2.2 Separation Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Voronoi Diagrams and Geometric Permutations for Pairwise Disjoint Discs . 14 3.2 Power Diagrams and Geometric Permutations for Pairwise Disjoint Discs . . 16 3.3 The Case of Pairwise Disjoint Unit Discs . . . . . . . . . . . . . . . . . . . . 17

4 Geometric Permutations in Higher Dimensions

4.1 Geometric Permutations of Pairwise Disjoint Balls in

5 Lower Bounds

IRd

. . . . . . . . . . . 27

5.1 Lower Bounds on Geometric Permutations for Disjoint Balls in

1

27

IRd

34

. . . . . . 34

Chapter 1 Introduction 1.1 Overview

Let A be a family of bodies in IRd. A line l is said to be transversal for A if it intersects every member of A. If A consists of pairwise disjoint convex bodies, then a line transversal for A induces a pair of linear orderings on A corresponding to its two possible orientations. Such pairs of linear orderings are called geometric permutations. Consider geometric permutations induced by line transversals to families A of pairwise disjoint convex sets in IRd. Let gd(n) be the maximum number of such permutations over all such families A of size n. The following asymptotic bounds are known on gd(n). 1. g2(n) = 2n ? 2 (see [10]). 2. gd (n) = (nd?1 ) (see [16]). 3. gd (n) = O(n2d?2 ) (see [21]). It is conjectured that g3 (n) is O(n2). In this thesis we introduce the following results: (i) We prove that the number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in IRd , is O(nd?1 ). This improve substantially the general upper bound of O(n2d?2 ) given in [21]. (ii) We also study the relation between geometric permutations and certain kinds of Voronoi diagrams or power diagrams induced by the given objects, and use it to show that the number of geometric permutations of a suciently large collection of pairwise disjoint unit discs in the plane is 2, improving a previous upper bound of 3 given in [15]. (iii) We establish sharp lower bounds, on the number of geometric permutations induced by line transversals to collections of pairwise disjoint balls in d-dimensional Euclidean space. The bounds are (nd?1 ), and match the known lower bounds of [16] (which are constructed 2

for other types of objects). Hence for families of n pairwise disjoint balls in IRd we have an asymptotically tight bound of (nd?1 ) on the maximum number of geometric permutations. This is the rst known sharp bound on the number of geometric permutations in IRd, for d > 2. We rst review the literature on transversals and geometric permutations. In Chapter 2 we introduce a few technical concepts that we will be using in subsequent chapters. In Chapter 3 we study geometric permutations in the plane. In particular we focus on geometric permutations induced by families of pairwise disjoint discs. We also study the relation between geometric permutations and Voronoi diagrams. In Chapters 4 and 5 we prove that the maximum possible number of geometric permutations of families of n pairwise disjoint balls in IRd is (nd?1).

1.2 k-Transversals

A k-transversal for a family A of convex bodies in IRd is an ane subspace of dimension k, called a k- at (such as a point, line, plane or hyperplane), which intersects every member of the family. The set of all k-transversals to A forms a topological space, denoted Tkd(A), lying in the \ane Grassmannian" of all k- ats in IRd (see [11] for more details on the properties of this space). The space of k-transversals to A is not necessarily connected, even for line transversals in the plane (i.e. T12). The space of line transversals to a family A of n pairwise disjoint compact convex sets in the plane can have at most 2n ? 2 connected components, and this bound can be attained (see [10, 11]). Much of the early work on transversals was motivated by Helly's theorem (see [6]) and directed toward giving `local' conditions for the existence of a k-transversal of A. A typical example is Hadwiger's transveral theorem, which states that a nite family A of pairwise disjoint convex sets in IR2, has a line transversal if and only if there exists a linear ordering of A such that every three members of A are intersected by a directed line in the given order. See [6] for a comprehensive review of these works. More recently, researchers have begun to study the structure of Tkd(A), the space of all k-transversals of A. Computer scientists, in particular, have been interested in explicitly constructing representations of this set and in analyzing its combinatorial complexity. The boundary of Tkd(A) consists of k-transversals which are tangent to one or more members of A. A face of this boundary is a maximally connected region of k- at transversals which are tangent to a xed subfamily of A. The combinatorial complexity of Tkd(A) is the number of such faces. A major problem has been to bound this combinatorial complexity for various families of convex sets, particularly for convex sets bounded by algebraic surfaces, and thus bound the size of an explicit representation of Tkd(A). 3

1.3 Hyperplane Transversals

Most of the progress has been made in bounding the complexity of Tdd?1(A), the space of hyperplane transversals to A. Let D() be the duality mapping which take each \nonvertical" hyperplane

h = f(x ; : : :; xd) : xd =  x +  x +  + d? xd? + dg: 1

1

1

2

2

1

1

to the point D(h) = (1; : : :; d) in dual space. The same duality mapping maps each point ( 1; : : :; d) 2 IRd to the hyperplane xd = ? 1x1 ? 2x2  ? d?1xd?1 + d. See [7] for details. The set of non-vertical hyperplane transversals to A is represented by a set of points in dual space. To construct this set in dual space, consider a single compact convex set a 2 A. The D-images of the non-vertical hyperplanes tangent to a form two unbounded surfaces in the dual space. The hyperplanes intersecting a are mapped to the points between these two surfaces. More precisely, de ne

a ( ; : : :; d? ) = maxfd : D? ( ; : : : ; d? ; d) \ a 6= ;g +

1

1

1

and

1

1

?a ( ; : : :; d? ) = minfd : D? ( ; : : :; d? ; d) \ a 6= ;g: For xed ( ; : : :; d? ) the set 1

1

1

1

1

1

1

fD? ( ; : : : ; d? ; d) : ?a ( ; : : : ; d? )  d  a ( ; : : : ; d? )g 1

1

1

1

1

+

1

1

is a set of parallel hyperplanes which intersect a. For each a 2 A, the graphs of the functions +a and ?a de ne two surfaces in IRd . The \corridor" of points between those surfaces represents the set of hyperplanes intersecting a. (If a is not closed, the boundaries of the corridors may not represent hyperplanes intersecting a. If a is not bounded, the functions +a and ?a may not be de ned at all points.) The intersection of all these corridors represents the set of non-vertical hyperplane transversals to A. Since each corridor is bounded by a \top" surface and a \bottom" surface, the points corresponding to hyperplane transversals are the points below all the \top" surfaces and above all the \bottom" surfaces. Formally, the space of hyperplane transversals to a family A of compact convex sets is

 ( ; : : :; d? )g: fD? ( ; : : : ; d? ; d) : max ?( ; : : : ; d? )  d  min a2A a a2A a 1

1

1

1

1

+

1

1

The function mina2A +a (1; : : :; d?1) is called the lower envelope of the functions f+a ga2A. Similarly, maxa2A ?a (1; : : : ; d?1) is the upper envelope of the functions f?a ga2A. A representation of the space of hyperplane transversals of A can be constructed by computing the 4

lower and upper envelopes of f+a g and f?a g, respectively, and then by intersecting these envelopes, or, more precisely, by computing the region enclosed between these envelopes. Upper and lower envelopes have been studied extensively over the past fteen years, particularly by Sharir and his colleagues, who made great progress in bounding their asymptotic combinatorial complexity. See the recent book [19] for a comprehensive presentation of this study. In particular Edelsbrunner, Guibas and Sharir applied this analysis to bound the combinatorial complexity of the space of hyperplane transversals to a family of convex polytopes:

Theorem 1.1 [8, 18] Let A be a family of n convex polytopes in IRd with a total of nf faces

(of all dimensions). The combinatorial complexity of the space of hyperplane transversals to A is O(ndf?1 (n)). Here (n) denotes the slowly growing inverse of the Ackermann function. Let s (n) denote the maximum length of an (n; s) Davenport-Schinzel sequence. (See [19] for the de nition and discussion of Davenport-Schinzel sequences and the Ackermann function and their relationship to upper and lower envelopes.) sFor any xed s, s(n) is ? almost linear but slightly superlinear in n, and is O(n (n)O( (n) )). Atallah and Bajaj [4] bounded the complexity of the space of line transversals in the plane in terms of s (n); see also [19]. (

2

2

)

Theorem 1.2 [4] Let A be a family of n compact connected sets in IR such that any two members of A have at most s common supporting lines. The combinatorial complexity of the space of line transversals to A is O(s (n)). 2

In IR3, Agarwal, Schwarzkopf and Sharir ([3]) bounded the complexity of the space of plane transversals to convex sets bounded by (picewise) algebraic surfaces with degree less than or equal to some xed constant.

Theorem 1.3 [3] Let A be a family of n convex sets in IR , each bounded by O(1) algebraic 3

surfaces of bounded degree. The combinatorial complexity of the space of plane transversals to A is O(n2+ ), for any  > 0, where the constant of proportionality depends on  and on the number and degree of the surfaces bounding each member of A. For hyperplane transversals to families of balls, the known bounds on the combinatorial complexity of the space of transversals is drastically lower.

Theorem 1.4 [13] Let A be a family of n (d ? 1)-balls in IRd. The combinatorial complexity of the space of hyperplane transversals to A is O(ndd= e). 2

It is not known if this bound is asymptotically tight in the worst case. 5

The proof by Houle et al. [13], is based on the fact that a convex polytope in IRd+1 which is the intersection of n half-spaces has O(ndd=2e) faces (this is the so-called upper bound theorem; see [17]). Represent each hyperplane

h = f(x ; : : :; xd) :  x +  x +  + dxd + d = 0g uniquely by the (d + 1)-tuple ( ; : : :; d ) where  +  + d = 1. (Note that this dual representation is di erent from the previous duality D().) The hyperplanes which intersect a ball centered at (c ; c ; : : : ; cd) with radius r are: f( ; : : : ; d ) : r  ( c +  + dcd + d )  ?r and  +  + d = 1g: Thus each ball de nes two linear inequalities or, equivalently, two half-spaces. The space of hyperplane transversals for n balls is thus the intersection of 2n half-spaces, two for each ball, and of the cylinder  +  + d = 1. The intersection of the 2n half-spaces has O(ndd= e) faces and the intersection of the resulting polytope with the cylinder has the same asymptotic upper bound on its complexity. 1

1

1

1

1

1

2

2

2

+1

2 1

+1

2

2

+1

1 1

2 1

+1

2 1

2

2

1.4 Spaces of Line Transversals The research in this thesis is focused on the study of spaces of line transversals. Such a space can also be represented as the region enclosed between an upper envelope and a lower envelope. For example, lines in IR3 can be represented by 4 real parameters. Speci cally, let us consider only lines not parallel to the yz-plane. Any such line l is uniquely represented by its projections on the xy and xz planes: y = 1x + 2, z = 3x + 4, so we can represent l by the point (1; 2; 3; 4) 2 IR4. Let a be a convex set in IR3. For xed 1; 2; 3, the set of lines (1; 2; 3; 4) that intersect a (if nonempty) is obtained by translating a line parallel to itself in the z-direction between two extreme values (1; 2; 3; ?a (1; 2; 3)) and (1; 2; 3; +a (1; 2; 3)), which represent lines tangent to a from below and from above, respectively. (Note that the functions ?a and +a are in general only partially de ned.) If we assume that the boundary of each object a 2 A is bounded by a xed number of algebraic surfaces of bounded degree, then the functions ?a ; +a also have `bounded algebraic complexity'. Hence, as in the case of hyperplane transversals, T13(A) can be represented as f(1; 2; 3; 4) : max ? ( ;  ;  ;  )  4  min +( ;  ;  )g: a2A a 1 2 3 4 a2A a 1 2 3 The problem is that there are no known sharp bounds on the complexity of a region enclosed between an upper envelope and a lower envelope in IR4. The complexity of a single envelope of n functions with `bounded algebraic complexity', in IR4, is O(n3+ ), for any  > 0 [19], and it is conjectured that a similar bound holds for the complexity of the above `sandwich region'. 6

In IRd, for d > 3, a similar situation arises. Lines in IRd can be represented by 2d ? 2 real parameters, and an appropriate choice of parameters maps T1d(A) to a region enclosed between an upper envelope and a lower envelope in IR2d?2. Again, no sharp bounds are known for the complexity of such a region. However, in spite of the lack of sharp (near-cubic) general upper bounds for the combinatorial complexity of the space of line transversals in IR3, such bounds have been obtained in some special cases. A series of papers have led to nearly tight bounds by Agarwal [2] on the complexity of the space of line transversals to convex polytopes in IR3:

Theorem 1.5 [2] Let A be a family of n convex polytopes in IR with a total of nf faces. The combinatorial complexity of T (A) is O(nf log n). 3

3 1

3

A very recent result by Agarwal, Aronov and Sharir [1] bounds the complexity of the space of line transversals to balls in IR3.

Theorem 1.6 [1] Let A be a family of n balls in IR . The combinatorial complexity of the space of line transversals to A is O(n  ) for any  > 0, and can be (n ) in the worst case. 3

3+

3

In higher dimensions, there are no published tight or nearly tight bounds on the combinatorial complexity of the space of line transversals. Finally we mention the case of Tkd(A), for 1 < k < d. This case is harder to analyze and has been less studied so far. k- ats in IRd can be represented by (k + 1)(d ? k) real parameters, and, as above, an appropriate choice of parameters maps Tkd(A) to a region enclosed between an upper envelope and a lower envelope in IR(k+1)(d?k). Needless to say, no sharp bounds are known for the complexity of such a region. Some recent nontrivial analysis of these spaces is given by Goodman et al. [12].

1.5 Geometric Permutations

In this thesis we are interested in particular in the case where the elements in A are pairwise disjoint. Let A be a family of pairwise disjoint convex sets in IRd. A directed line transversal intersects the elements of A in a xed order. An undirected line induces a pair of linear orderings or \permutations" on A corresponding to its two possible orientations. Katchalski et al. [14] were the rst to study such pairs of orderings and called them geometric permutations. If two line transversals induce di erent geometric permutations on A, then they must lie in di erent connected components of T1d(A). In section 2.1 we prove a stronger version of this claim which implies that if two lines induce di erent geometric permutations on A, then they must lie in di erent connected components of the orientation space of A, which is the space of all orientations of line transversals to A. Thus each connected component of this space is associated with, at most, a single geometric permutation of A. In the plane, the converse is also true. 7

Theorem 1.7 [20] Two lines that induce the same geometric permutation on a nite family A of pairwise disjoint compact convex bodies in IR must lie in the same connected component in T (A). Proof: Consider two lines l ; l that induce the same geometric permutation on a set A of 2

2 1

1

2

pairwise disjoint convex bodies in the plane. It is easy to see that if l1 and l2 are parallel, then we can move l1 continuously towards l2, keeping it parallel to l2, until it coincides with l2, such that the moving line is always transversal to A. Otherwise, the two lines inetersect and divide the plane into four quadrants. For each a 2 A choose two points xa 2 l1 \ a and ya 2 l2 \ a and consider the segments xaya, for a 2 A. These segments are contained in the respective members of A, i.e., xaya  a, 8a 2 A. It is easy to see that all these segments are contained in a xed pair of opposite quadrants except, maybe, for at most one segment a0. If all segments are contained in two opposite quadrants, we can rotate the line l2 around the point of intersection with l1 until it coincides with l1, such that the rotated line keeps intersecting all these segments. If we have a segment a0 contained in a third quadrant, then it is easy to see that the endpoints of a0 are nearest to l1 \ l2 along both lines l1; l2. Hence, we can rotate the line l2 around l2 \ a0 until it coincides with the line la0 through a0 and then rotate la0 around the point la0 \ l1 until it coincides with l1 (see Figure 1.1), so that all intermediate lines intersect all the segments. In IR3, this is no longer true. Two lines that induce the same geometric permutation on a family A in IR3 may lie in di erent connected components in T13(A); see [11] for an example. In 1985, Katchalski, Lewis and Zaks [15] constructed families of n  4 pairwise disjoint convex sets in IR2 which have 2n ? 2 geometric permutations (see Figure 1.2). Five years later, Edelsbruner and Sharir [10] showed that 2n ? 2 is the maximum possible number of geometric permutationms in the plane. Consider geometric permutations induced by line transversals to families of pairwise disjoint compact convex sets in IRd. Let gd (n) be the maximum number of such permutations over all such families A of size n. In 1992 Katchalski et al. [16] generalized their lower bound construction and showed that there exists families A of n pairwise disjoint convex sets in IRd admitting (nd?1) geometric permutations. In this work we show that this lower bound is true even for collections of balls in d dimensions. The only known general upper bound for the number of geometric permuations of collections of pairwise disjoint convex sets in IRd is O(n2d?2), and is due to Wenger [21]. Even for line transversals in IR3, the known asymptotic upper and lower bounds, O(n4) and (n2), still have a wide gap. In this thesis we close this gap for families of balls and for some other families. We summarize the known general results on geometric permutations: 1. g2(n) = 2n ? 2 (see [10]). 8

l2

la

0

la \ l 1

la \ l 2 0

0

a

0

l1

Figure 1.1: Two lines that induce the same geometric permutation must lie in the same connected component.

9

n ? 2 segments

1 n?2

n?1

n

(n ? 1; 1; 2; : : :; n ? 2; n) (1; 2; : : :; i; n ? 1; i + 1; : : :; n)

(1; n; 2; : : :; n ? 2; n ? 1)

Figure 1.2: An example of n convex sets admitting 2n ? 2 geometric permutations.

10

2. gd (n) = (nd?1 ) (see [16]). 3. gd (n) = O(n2d?2 ) (see [21]). It is conjectured that g3(n) is O(n2 ). Note that Theorems 1.5 and 1.6 imply near-cubic upper bounds for g3(n) for families of convex polytopes or of balls.

11

Chapter 2 Geometric Preliminaries In this chapter we develop a few technical concepts that we will be using in subsequent chapters. We also generalize the notion of separation set used by Wenger (see [21]).

2.1 Geometric Permutations and Orientations Space Let S be a collection of pairwise disjoint convex sets in IRd, and let l1; l2 be two directed line transversals to S . Denote by l : S ?! f1; 2; : : : ; ng the permutation induced by a line l. For simplicity, denote by 1 (resp. 2) the permutation l (resp. l ). We have: 1

2

Claim 2.1 l k l implies  =  . 1

2

1

2

Proof: Suppose, to the contrary, that 1 6= 2. This means that there are at least two sets si; sj 2 S such that 1(si) < 1(sj ) and 2(si) > 2(sj ). Now choose points p1i 2 l1 \ si, p2i 2 l2 \ si, p1j 2 l1 \ sj , p2j 2 l2 \ sj . It is easy to see that the two segments segi = pi1pi 2 and segj = pj 1pj 2 are contained in the same plane that passes through l1 and l2 and must intersect each other. Moreover the sets si; sj are convex so segi  si and segj  sj holds. Hence segi \ segj 6= ; implies that si \ sj 6= ;, contradicting their disjointness. Given an orientation in IRd, all line transversals l, to S with orientation induce the same permutation l. We can associate every orientation in IRd with a unique point on the unit sphere S d?1. Let l be an oriented line transversal to S . Denote by (l) the point on S d?1 that corresponds to the orientation of l. Let (S ) = f (l)j l is transversal to S g, let C (S ) denote the number of connected components of (S ), and let GP (S ) denote the number of di erent geometric permutations of S .

Claim 2.2 GP (S )  C (S ).

Proof: We show that if l1 and l2 are two line transversals to S such that l 6= l , then (l1) and (l2) must lie in di erent connected components of (S ). Indeed, there are two 1

12

2

sets si and sj that appear in di erent order in 1 and 2. Choose a hyperplane H strictly separating si and sj . The space of all orientations of lines that are parallel to H corresponds to a great circle on the unit sphere and no line with orientation on this circle is transversal to S . Moreover, it is easy to see that the great circle divides the unit sphere into two connected components C1 and C2 such that (l1) 2 C1 and (l2) 2 C2. Hence (l1) and (l2) lie in di erent connected components of (S ). Claim 2.3 In IR2 we have GP (S ) = 12 C (S ), provided that jS j  2. Proof: The proof of Claim 2.2 implies that each component of (S ) induces exactly one geometric permutation. Theorem 1.7 implies that each geometric permutation arises in exactly one component of T12(S ) and thus in a single pair of antipodal components of (S ). (Since jS j  2, the components of (S ) must come in antipodal pairs.) These observations imply the asserted claim.

2.2 Separation Sets

De nition 2.4 Let dS be a family of pairwise disjoint convex sets in IRd , and let P be a set

of hyperplanes in IR passing through the origin. We say that P is a separation set for S if for each pair si; sj 2 S there exists a hyperplane H such that si and sj are contained in di erent closed half-spaces bounded by H and such that H is parallel to a hyperplane in P .

Lemma 2.5 Let S be a collection of pairwise disjoint convex sets in IRd and let P be a d ? separation set for S . Then GP (S ) = O(jP j ). Proof: Consider the arrangement, A(P ), of the great spheres in S d? associated with the 1

1

hyperplanes in P (i.e. each of these great spheres consists of all orientations parallel to some hyperplane in P ). The arrangement partitions the unit sphere into O(jP jd?1) connected components. Fix a connected component C of (S ) and a pair of sets si; sj 2 S . There exists a hyperplane H , that separates them and corresponds to one of the great spheres in the arrangement A(P ), such that C lies in a xed side of this sphere. This means that every oriented line that intersects both si and sj with orientation in C , must intersect the sets in a xed order. This is true for every pair of sets in S . Therefore every line transversal to S with orientation in C induces a xed order on S . Thus the number of geometric permutations for S is smaller than or equal to the number of connected components in the arrangement A(P ), and so implies the asserted bound. Corollary 2.6 gd (n) = O(n2d?2 ). d Proof:  For any set S of n pairwise disjoint convex sets in IR there exists a separation set of size n2 : separate each pair of sets in S by a di erent hyperplane. The corollary is now immediate from Lemma 2.5. 13

Chapter 3 Geometric Permutations in The Plane 3.1 Voronoi Diagrams and Geometric Permutations for Pairwise Disjoint Discs Let A be a family of n pairwise disjoint unit discs in the plane, let l be a line transversal to A and let  = (1; : : :; n) be the permutation of the discs in A, induced by l. Now consider the Voronoi diagram of the n centers of the discs. Note that each disc is contained in the interior of the Voronoi cell of its center, therefore l intersects the interiors of all cells in the above Voronoi diagram. Moreover, the line l intersects these Voronoi cells in the same order as the corresponding discs and therefore induces the same permutation  on the Voronoi cells. We therefore conclude that an upper bound on the number of geometric permutations for Voronoi diagrams of n points in the plane also serves as an upper bound on the number of geometric permutations for n pairwise disjoint unit discs. A natural and interesting question thus arises: Given a set S of n points in the plane, what is the maximum number of permutations, induced by line transversals to the interiors of the cells of the Voronoi diagram, Vor (S ), of S . Let S be a set of n points in the plane. Denote by GPV (S ) the maximum number of possible geometric permutations of the cells of Vor (S ) and let GPV (n) = maxjSj=n GPV (S ).

Theorem 3.1 GPV (n) = 2n ? 2.

Proof: The cells of Vor (S ) form a collection of convex sets in the plane with pairwise disjoint interiors. A simple adaptation of the analysis of [10] implies that GPV (n)  2n ? 2. In order to show that GPV (n)  2n ? 2, consider the following construction. Put pn = (1; 0) and pn?1 = (?1; 0). Put n ? 2 additional points, p1; : : :; pn?2 , on the positive y-axis suciently close to one another such that pi lies below pi?1 , for i 2 f2; : : : ; n ? 2g, and such that the perpendicular bisector of p1 pn intersects pn?2 pn . It is easy to see that the Voronoi diagram of this set of points admits 2n ? 2 permutations of the form:

14

(1; 2; 3; n; :::; n ? 1) 1 2 n-2 n

n-1

(1; n; 2; :::; n ? 1)

Figure 3.1: A Voronoi diagram of n points admitting 2n ? 2 geometric permutations

15

 (1; 2; : : : ; i ? 1; n; i; i + 1; : : : ; n ? 2; n ? 1)  (1; 2; : : : ; i ? 1; n ? 1; i; i + 1; : : : ; n ? 2; n) for i 2 f0; : : : ; n ? 2g. See Figure 3.1 for an illustration. Note that this construction is

similar to that of [11], and can serve as an alternative lower bound construction, showing that g2(n)  2n ? 2. However, this bound is not tight, when we replace the Voronoi cells by the original family of pairwise disjoint unit discs. This is a special case of families of disjoint translates of a convex set. Katchalski et al. [15] proved that for such a family of disjoint translates the maximum number of geometric permutations is 3. In Section 3.3 we improve this bound to 2 for the case of pairwise disjoint unit discs (see Theorem 3.7 for a precise statement). Meanwhile, as a digression, we consider the case of noncongruent discs, and provide, by using their power diagram [5], an alternative and simpler proof that the number of their geometric permutations is O(n).

3.2 Power Diagrams and Geometric Permutations for Pairwise Disjoint Discs

The power pow(x; s) of a point x with respect to a ball s in IRd is given by d2(x; z) ? r2, where d denotes the Euclidean distance and z and r are the center and the radius of s, respectively. The power diagram of a nite set S of balls in IRd (denoted by PD(S )) is a ployhedral cell complex that associates each s 2 S with the convex polyhedral domain fx 2 IRdjpow(x; s)  pow(x; t) 8t 2 S g. For more details about power diagrams see [5]. We use the following known properties about power diagrams.

Lemma 3.2 Let S be a set of n  3 pairwise disjoint discs in the plane.  PD(S ) consists of exactly n cells, such that each disc is contained in the interior of a unique cell.

 PD(S ) consists of at most 3n ? 6 edges. The set H of lines that pass through the origin and are parallel to the edges of PD(S ) is a separation set for S , of size  3n ? 6. This follows from the fact that any pair of disjoint polygons are separated by a line that passes through an edge of one of them. We immediately get:

Corollary 3.3 The number of geometric permutations for a collection of n pairwise disjoint discs in IR is at most 3n ? 6. 2

16

Edelsbrunner et al. [9] proved that a collection of n pairwise disjoint compact convex sets in the plane can be covered with non-overlapping convex polygons with a total of at most 6n ? 9 sides and 3n ? 6 distinct slopes. This, in turn, provides another proof of the linear bound, for general pairwise disjoint compact convex sets, however, their proof is somewhat more involved (and, of course, the bound is weaker than that of [10] in both proofs).

3.3 The Case of Pairwise Disjoint Unit Discs In this section we study the case of families of pairwise disjoint unit discs in the plane. We show that for n suciently large, such a family admits at most two geometric permutations. The case of pairwise disjoint unit discs is a special case of disjoint translates of a convex body. Geometric permutations of disjoint translates of a convex set were studied by Katchalski et al. in [15, 16] where the following result was proved.

Theorem 3.4 The maximum number of geometric permutations for families of n disjoint

translates of a convex body in IR2 is 3. We omit the proof here. For a construction that achieves this bound (for unit discs) see Figure 3.2. For an example of a family of more than three translates admitting 3 geometric permutations see Figure 3.3. This construction can be generalized to any number of translates, of the same body. However, we show that if the convex body s is a disc, there exists a constant c such that for any set S of more than c pairwise disjoint translates of s, the number of geometric permutations for S is at most 2. We begin the proof with some observations:

Lemma 3.5 Let S be a collection of n pairwise disjoint unit discs in the plane, such that

S admits at least one line transversal. There exists two discs such that the distance between their centers is at least n=4 ? 2. Proof: If l is a line transversal to S then all discs in S are contained in a strip of width 4 around l. Denote by d the distance between the center of the rst disc stabbed by l and the center of the last disc stabbed by l. Since the sum of areas of the discs is less than the area of the rectangle bounding the discs in direction l, we have: (d + 2)  4  n which implies the above inequality.

Claim 3.6 Let a and b be two disjoint unit discs in the plane. Then every line transversal

to a and b must cross both wedges that contain a and b of the double wedge de ned by the two common separating tangents of a and b. Therefore the orientation of each such line transversal must belong to the appropriate pair of antipodal intervals of orientations de ned by the two tangents. See Figure 3.4 for an illustration.

17

1

2

3

Figure 3.2: An example of three unit discs admitting three geometric permutations

18

Figure 3.3: An example of ve translates admitting three geometric permutations.

b

a

l

Figure 3.4: A transversal l for two discs must cross the double wedge de ned by the two common tangents.

19

As a result of Lemma 3.5 and Claim 3.6, and using the fact that the angle between the two separating tangents is proportional to arcsin( d1 ) where d is the distance between the discs, we can assume that all orientations of line transversals to a collection of pairwise disjoint unit discs lie in an -neighborhood (more precisely, in the union of two antipodal -neighbourhoods) on the unit circle of directions, for a small  to be xed later. (Note that  = O( n1 ), so we can make  as small as we wish by increasing the number of discs.) We proceed by testing the number of connected components of the orientations space (S ) to a family S of disjoint unit discs in IR2. Let  be the the angle between the two separating common tangents of the farthest pair of discs in S . All line transversal orientations should lie in the union I of the two antipodal -orientations mentioned above. Moreover, a line transversal to S is, in particular, a transversal to all pairs of discs in S and therefore must lie in the intersection of all the corresponding unions of antipodal intervals, one pair for each pair of discs in S . The only case where each of the -neighborhoods in I can be broken into more than one connected component is when a pair of discs are close enough such that their common tangents de ne an antipodal pair of intervals of admissible orientations whose sizes are close to  and such that they both cross the two antipodal intervals in I . See Figure 3.5 for an illustration. In that case, simple trigonometry shows that the two corresponding discs must be very close to one another (the distance between their centers is at most 2 + O(2) = 2 + O( n1 ) ), and that the two discs in such a pair must be adjacent in any possible geometric permutation. Moreover, the line connecting the centers of such a pair of discs must be almost perpendicular to any line transversal of S . This implies that a disc can belong to at most one such pair. Also, pairs of discs that are not as close to each other appear in the same order in any geometric permutation. We show next that at most one pair of discs can swap in any geometric permutation. Assume to the contrary that there are two pairs (a; b) and (c; d) that consist of four distinct discs such that there are at least three geometric permutations in which, without loss of generality, both pairs appear in the following possible orders: 1. (a; b; c; d) 2. (b; a; c; d) 3. (a; b; d; c) (Note that we may assume that there are three distinct geometric permutations, for otherwise the theorem we want to prove holds. We thus do not have to consider the case where both pairs are swapped in the same permutation.) We now turn to use the fact that any such permutation is a geometric permutation of the cells of the Voronoi diagram, de ned by the corresponding four centers. We consider all possible con gurations of these four centers. Case 1: The four centers are not in convex position. The two possible situations are that the centers are collinear (Figure 3.6) or one of the points is in the interior of the convex hull 2

20

I

I

Figure 3.5: Intersection of 2 pairs of antipodal arcs can break each interval of I into two connected components.

21

of the three others (Figure 3.7). In the rst situation it is easy to see that there is only one possible permutation for the four points. In the second situation, since one face of the Voronoi diagram is bounded, it cannot appear neither at the end nor at the beginning of any permutation, which contradicts the fact that each center is extreme in at least one of the above three permutations.

Figure 3.6: A Voronoi diagram of 4 collinear points admits only 1 geometric permutation Case 2: The points are in convex position. Here again we have two situations. The rst possible situation is when the four centers are cocircular (Figure 3.8). In this case there are at most two possible permutations, since any line transversal must cross three out of the four Voronoi edges and for each triple of edges, say (1; 2; 3), crossed by a line, there is no line crossing the triple (1; 4; 3) because the angle of the sector 143 is larger than  (see Figure 3.8). Hence there are at most two such triples of edges, which induce at most two geometric permutations. The second situation is when the centers are not cocircular. The Voronoi diagram then consists of four unbounded cells, with a unique pair of nonadjacent cells. In the three permutations that are assumed to exist, a and d are the only pair that are never adjacent, which implies that the nonadjacent cells of the diagram must be the cells of a and of d, as shown in gure 3.9. We inspect the Voronoi edges eab and ecd. Denote by lab the perpendicular

22

Figure 3.7: A Voronoi diagram of 4 points with one point in the interior of the convex hull of the 4 points.

23

2

1 a

c

b

3

d



>

4

Figure 3.8: A Voronoi diagram of four cocircular points

d b a

c

Figure 3.9: A Voronoi diagram of 4 points not cocircular but in convex position 24

bisector of segment ab, i.e., the line passing through edge eab. We consider all possible relative con gurations of the edges eab; ecd, as follows:

lcd

d b a

c lab

Figure 3.10: The lines lab and lcd are parallel Case (a): lab and lcd are parallel (see Figure 3.10). In that case lab separates a from b and d. Thus no line induces the permutation (b; a; c; d), a contradiction. Case (b): lab intersects ecd (see Figure 3.11). In that case lcd separates d from a and c. Hence (a; b; d; c) is not a possible geometric permutation, a contradiction. Case (c): lcd intersects eab (see Figure 3.12). In that case lab separates a from b and d and then again it is impossible to get the permutation (b; a; c; d). These are all the possible con gurations of the given four points, and they all lead to a contradiction. Hence there is a unique pair of discs that can change its order in the geometric permutations. We have proved:

Theorem 3.7 There exists a constant c such that any family of more than c pairwise disjoint congruent discs in the plane admits at most two geometric permutations.

Remark 3.8 As we will show in Section 5.1, this result does not hold without the bounded ratio assumption.

25

lcd

d b

eab

ecd

a

c lab

Figure 3.11: The line lab intersects Voronoi edge ecd

d b lcd

ecd

eab

c

a lab

Figure 3.12: The line lcd intersects Voronoi edge eab 26

Chapter 4 Geometric Permutations in Higher Dimensions In this chapter we prove that a set of n pairwise disjoint balls in IRd admits at most O(nd?1 ) geometric permutations. In Section 5.1 we show that there exist sets of n pairwise disjoint balls in IRd that admit (nd?1) geometric permutations, for arbitrarily large n. Hence our bound is tight in the worst case.

4.1 Geometric Permutations of Pairwise Disjoint Balls in IRd

Let S be a given set of n pairwise disjoint balls in IRd. The main step of the proof is to show that S admits a separation set of size O(n). As a matter of fact, we prove the stronger result that there exists a set P of O(n) hyperplanes such that each pair of balls in S is separated by a hyperplane in P . We rst introduce the following de nitions, for a coordinate frame F , and for a ball b 2 S .

 rb denotes the radius of b and pb denotes its center.  BOXb F denotes the axis-parallel bounding box of b in F .  AbF denotes the orthant that contains b in the axis-parallel coordinate frame (relative ( )

( )

to F ) whose origin is at the vertex v of BOXb(F ) with largest coordinates. The coordinate hyperplanes in this frame pass through the d facets of BOXb(F ) that are adjacent to v.  Ob(F ) denotes the opposite orthant of A(bF ). 27

 Ib denotes the set of all balls in S , with radius larger than or equal to rb and intersecting

the ball b that is cocentric with b and has radius 10d rb. Since the balls are pairwise disjoint, the size of Ib is O(1) for every b 2 S . We will use a constant number of coordinate frames, each rotated about the origin, so that the following property holds: For any pair of balls a; b 2 S there exists a frame F such that the line that connects pa with pb , and oriented from the former to the latter, points into the positive orthant of F and forms an angle of at least d with each coordinate hyperplane, for some suciently small parameter d > 0 (that depends on d). It is easily veri ed that such a collection of O(1) frames exist, provided that d is small enough (by the isoperimetric q log d inequality of [], we may choose d = 0:5 d ). Let a(b) denote the orientation of this line and let F denote such a collection of O(1) frames. We construct a separation set P in the following three stages: 1. For each ordered pair of balls (b; c) such that b 2 S and c 2 Ib we take a hyperplane b;c that separates b and c. Denote the set of these hyperplanes by P1. Clearly, the size of P1 is O(n). 2. For each frame F 2 F and for each ball b 2 S , take the 2d axis-parallel hyperplanes (F ) supporting BOXb and add them to P . Let P2 denote the set of all these hyperplanes. Clearly jP2j = O(n). 3. Let F 2 F and b 2 S . Consider the set Sb(F ) of all balls in S that cut the d hyperplanes de ning A(bF ) and whose centers lie in Ob(F ). For a ball c in S we say that b 9dr : 10 d ? d(rb+rc )  10d rb+rc ) rc ( d?1)  rb(10d ? d) ) rc  ( p b b d?1 b

p

IRd

3 2

p

3 2

3 2

Consider a partition A of the positive orthant, S +, of the unit sphere S of directions in into O(1) regions, each of diameter at most 2pdd .

Lemma 4.1 Let a; b; c 2 S and let F 2 F . Suppose that b; c 2 S^aF , and that a(b); a(c), ( )

when regarded as orientations in F , lie in the same region of A, and outside the d bands of width d , centered at each of the great (d ? 2)-spheres on S that lie in the coordinate hyperplanes. Then all the components of the vector from pb to pc are positive, or all are negative. Moreover, in the former case the triangle 4papb pc is obtuse at pb and in the latter case the triangle is obtuse at pc .

28

pc y

z

pb

x

pa

Figure 4.1: The proof of Lemma 4.1. Proof: Assume without loss of generality that rb  rc . Consider the triangle 4papb pc . Let = 6 pc pa pb, = 6 papc pb and = 6 papb pc . Put x = jpapb j and y = jpbpc j (see Figure 4.1 for an illustration). By the sine theorem we have:

x = y ) sin = x sin : sin sin y

p

The fact that a rb + rc. Hence we have

p

p

p

 d(ra + rb) sin  d(ra + rb) sin  d( d + 1) sin < pd sin : sin = x sin y rb + rc 2rb 2 The p fact that a(b) and a(c) lie in the same region of A implies that  pd . Hence, sin < d sin pd <  < sin d (which holds for our choice of d). Hence, either < d or >  ? d. We conclude that either the orientation of the line p??b! pc or that of ? p?c! pb di ers ? ? ! by at most d from the orientation of the line pa pc and therefore lies in the correct orthant, provided (as assumed in the lemma) that the region of A containing a(b) and a(c) is outside the d bands of width d around the boundary of S . Finally, the preceding analysis clearly implies the last assertion of the lemma. For a frame F 2 F and for a ball b 2 S , let S~bF denote the subset of S^bF consisting of balls c such that b(c) lies outside the bands de ned in Lemma 4.1. (Recall that these balls satisfy (i) c 2= Ib and (ii) rc  9drb.) For a pair a; b of balls, de ne a 1 ? 1; 1? 9d 8d 9d d as is easily veri ed. This contradiction completes the proof of the lemma. Theorem 4.3 Let P be the set of hyperplanes as constructed above. Any pair of balls in S are separated by a hyperplane in P . Proof: Let (b ; b ; : : :; bn) be a permutation of S for which the corresponding sequence of radii is nonincreasing. We will prove, by induction on i, that any pair of distinct balls in Si = fb ; b ; : : :; big is separated by a hyperplane in P . The claim is vacuously true for i = 1. Suppose it is true for some i < n, and consider Si . We thus have to show that bi is separated from all the balls in Si by hyperplanes in P . Let a be a ball in Si. If a 2 Ib , then, by construction, bi and a are separated by a hyperplane in P . Otherwise, there exists a frame F in which b (a) lies in the positive orthant and forms an angle of at least d with each of the coordinate hyperplanes. We refer to this property by saying that b (a) is centered in F . If bi and a are separated by a hyperplane parallel to one of the coordinate hyperplanes of F and supporting the vertex of BOXb F with maximal coordinates (a hyperplane belonging to P by construction) then we are done. Otherwise, we claim that bi