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Discrete & Computational Geometry

, Volume 48, Issue 1, pp 1–18 | Cite as

On Levels in Arrangements of Surfaces in Three Dimensions

  • Timothy M. Chan
Article

Abstract

A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n “pseudo-planes” or “pseudo-spherical patches” (where the main criterion is that each triple of surfaces has at most two common intersections), we prove that there are at most O(n 2.997) vertices at any given level.

Keywords

k-Level Arrangements The k-set problem 

1 Introduction

Given an arrangement of n surfaces in ℝ d , the level of a point p∈ℝ d is the number of surfaces strictly below p. Combinatorial and computational geometers have been baffled by the following simple, basic question:

Consider the number of vertices in the arrangement that have level equal to k. How large can this number be, asymptotically as a function of n and k?

This question has relevance to the analysis of algorithms for a number of fundamental geometric problems [14, 20, 22, 24].
Here is a summary of what is known:
  • The most famous case—dually related to the so-called k-set problem —concerns lines in 2-d. The early papers by Lovász [18] and Erdős et al. [16] showed that every arrangement of n lines has at most \(O(n\sqrt{k})\) vertices at level k, and furthermore, there exist arrangements of lines with Ω(nlogk) such vertices. Major improvements did not come for more than twenty years, until Dey [12] and Tóth [28] improved the upper bound to O(nk 1/3) and the lower bound to \(n2^{\varOmega(\sqrt{\log k})}\).

  • For planes in 3-d, Bárány, Füredi, and Lovász [5] were to first to obtain a nontrivial, subcubic upper bound of O(n 3−1/343)=O(n 2.9971). After a series of improvements (in chronological order [1, 4, 13, 15]), Sharir, Smorodinsky, and Tardos [25] gave the current best upper bound of O(nk 3/2). The current lower bound is \(nk2^{\varOmega(\sqrt{\log k})}\) [28].

  • For planes in 4-d, Sharir [23] obtained the most recent upper bound O(n 4−1/18) (improving a previous result by Matoušek et al. [21]). By a technique of Agarwal et al. [1], the bound can be made sensitive to k, namely, O(n 2 k 2−1/18).

  • For hyperplanes in a fixed dimension d>3, Bárány et al.’s proof in combination with the multicolored Tverberg theorem [6, 29] yielded an upper bound of \(O(n^{d-\alpha _{d}})\) for a very small α d =1/(2d) d . As Agarwal et al. [1] observed, the bound can be made sensitive to k, namely, \(O(n^{ \lfloor{d/2} \rfloor}k^{{ \lceil{d/2} \rceil}-\alpha_{d}})\).

  • The more general case of nonlinear curves in 2-d has also been studied extensively. Previous proofs for arrangements of lines, including Dey’s O(nk 1/3) result, can be adapted to arrangements of pseudo-lines , where each pair of curves intersects at most once [26]. For more general families, however, new techniques are required. Tamaki and Tokuyama [27] were the first to suggest the approach of “cutting” curves and obtained an O(n 23/12) upper bound on the number of vertices of any level in an arrangement of n pseudo-parabolas , where each pair intersects at most twice. The bound for pseudo-parabolas has eventually been reduced to O(nk 1/2logk), after a series of improvements ([7, 2, 8, 19, 10] in chronological order). In 2005, the author [8] proposed a new, simple approach that yielded an \(\tilde{O}(nk^{1-1/2s})\) upper bound for general curve families, where each pair intersects at most a constant s number of times, and the \(\tilde{O}\) notation hides small, inverse-Ackermann-like factors; further improvements were also given for even values of s and for specific curve families (for example, graphs of fixed-degree polynomials in one variable). A follow-up paper [10] obtained yet more improvements.

  • For polyhedral surfaces comprising O(n) triangles in 3-d, Agarwal et al. [1] gave an \(\tilde{O}(n^{2}k^{7/9})\) upper bound, which was improved to \(\tilde{O}(n^{2}k^{2/3})\) by Katoh and Tokuyama [17].

In this paper, we give upper bounds for nonlinear surfaces in 3-d. The surface families that can be handled by our proof are fairly general, and include a certain definition of pseudo-planes and pseudo-triangles , where the main condition is that each triple of surfaces intersects at most once, and pseudo-spherical patches , where the main condition is that each triple of surfaces intersects at most twice (see Sect. 2.3 for the precise general requirement and specific definitions used). For these particular surface families, our upper bound is O(n 3−1/286.97)=O(n 2.9966), which can be made k-sensitive, namely, O(nk 1.9966) for pseudo-planes and \(\tilde{O}(n^{2}k^{0.9966})\) for the other families, by Agarwal et al.’s observations [1]. It should be emphasized that while this barely subcubic bound may not look very impressive, it is the first demonstration that a nontrivial result is possible, and with any luck, improvements might subsequently follow.1 (The reader is asked to take a historical perspective and compare our result with Bárány et al.’s for planes [5].) Rather than the specific bound, the proof technique itself should be regarded as the main contribution.

Why can’t previous techniques be adapted to handle general surfaces? The existing upper-bound proofs for planes [4, 5, 13, 15, 25] and triangles [1, 17] in 3-d are all similar in that they are based (in part) on Lovász’s original approach [18]. It is not unthinkable that this approach could work for pseudo-planes, although finding an appropriate extension of Lovász’s main lemma was posed by Agarwal et al. [1] some time ago and is still unanswered. In any case, an extension of this approach to more general surfaces in 3-d (or even general curves in 2-d) is less imaginable due to a variety of obstacles (for starters, Lovász’s lemma is typically applied to points in dual space, but no point-surface duality is known). We therefore need a proof that is completely different from all previous proofs for planes. The author’s new technique for curves in 2-d [8] turns out to be just what is needed, but the extension to 3-d is not easy and requires a number of additional ideas, together with an intricate charging argument.

2 The Proof Plan

We may assume (by perturbation arguments) that the given arrangement is nondegenerate. To make it easier to understand, we first describe our proof plan for the case of pseudo-planes, which we formally define as surfaces satisfying the following requirement:

The surfaces are graphs of total bivariate functions such that (i) the intersection of each pair is an x-monotone curve, and (ii) the intersection of each triple is a single point.

Note that a more common definition of pseudo-planes would omit condition (i). Later in Sect. 2.3, we will consider generalizations that would relax both conditions and allow for certain non-pseudo-plane families.

2.1 The Previous Approach

As mentioned, the main approach is from the author’s previous paper [8]. The idea is to extend the problem by looking at the number of vertices at nearby levels. A nontrivial upper bound is then obtained by solving a recurrence/difference equation.

Given an arrangement of n pseudo-planes in ℝ3 and an integer k, let t i be the number of vertices in the arrangement with levels in the range (ki,k+i). Let Δt i =t i+1t i be the number of vertices with level ki or k+i. Our problem is to bound t 1, the number of vertices in the arrangement at level k. (A more common version of the problem is to bound the combinatorial complexity of the k-level , defined as the boundary of the set of all points in ℝ3 with level at most k; the two versions are known to be asymptotically equivalent—e.g., see [3].)

In the previous proof in 2-d [8], we bound t i in terms of Δt i by a simple charging argument, yielding an inequality of the form
$$ t_i \le c_0i\Delta t_i + \mbox{[some overhead term].} $$
(1)
(In the pseudo-line case, c 0=2.) With the base case t n =O(n 2), this recurrence gives \(t_{i}=O(n^{2-1/c_{0}}i^{1/c_{0}})\), implying a subquadratic bound for t 1.

We would like to derive a similar inequality in 3-d (this time, with t n =O(n 3)). Unfortunately, there seems no obvious way to make the original charging argument work in 3-d. Further ideas are needed ….

2.2 Reduction to 2-d

As it turns out, the difficulties arising in 3-d can be resolved by not working in 3-d at all. More precisely, we consider the 2-d subarrangement inside each of the n given surfaces, prove a general inequality in 2-d, and then “sum up” to get the 3-d inequality. (At least one previous proof, by Sharir et al. [25], was also in part based on summing up contributions from various 2-d subproblems, though in a very different way; our idea here was indirectly inspired by Sharir et al.’s proof.)

The subarrangement within a given surface σ is a pseudo-line arrangement when projected to the xy-plane. In studying this subarrangement, we need to classify each pseudo-line γ as one of two types, “red” or “blue”, depending on whether within σ, points above γ in the y-direction are above or below the surface defining γ in the z-direction.

We now need to prove an inequality in 2-d that more generally applies to a bichromatic arrangement.

Definition 2.1

Consider an arrangement of pseudo-lines in ℝ2, where each curve is colored red or blue. For a red curve γ, we say that a point p violates γ if p is strictly above γ; for a blue curve γ, p violates γ if p is strictly below γ. Define the level of p to be the number of curves violated by p. Call an intersection of two curves a monochromatic vertex if the two curves have the same color; call it a bichromatic vertex otherwise. Call a vertex at level in the range (ki,k+i) an interior vertex, and a vertex at level ki or k+i a boundary vertex.

Let \(t^{\mathrm{mo}}_{i}\) and \(t^{\mathrm{bi}}_{i}\) be the number of interior monochromatic and bichromatic vertices, respectively, and \(\Delta t^{\mathrm{mo}}_{i}=t^{\mathrm{mo}}_{i+1}-t^{\mathrm{mo}}_{i}\) and \(\Delta t^{\mathrm{bi}}_{i}=t^{\mathrm{bi}}_{i+1}-t^{\mathrm {bi}}_{i}\) be the number of boundary monochromatic and bichromatic vertices, respectively.

We would like to bound the number of interior vertices (\(t^{\mathrm{mo}}_{i}\) or \(t^{\mathrm{bi}}_{i}\)) in terms of the number of boundary vertices (\(\Delta t^{\mathrm{mo}}_{i}\) or \(\Delta t^{\mathrm{bi}}_{i}\)), as in (1). Unfortunately, this is not possible in the bichromatic setting, because as Fig. 1 indicates, both \(t^{\mathrm{mo}}_{i}\) and \(t^{\mathrm{bi}}_{i}\) can be quadratic in the worst case.
Fig. 1

A “bad” example of a bichromatic arrangement, with all vertices shown having the same level. In all figures, unbold lines are red and bold lines are blue

To overcome this problem, we need yet another idea: charge interior monochromatic vertices (\(t^{\mathrm{mo}}_{i}\)) not only to boundary vertices (\(\Delta t^{\mathrm{mo}}_{i}\) and \(\Delta t^{\mathrm{bi}}_{i}\)) but also to interior bichromatic vertices (\(t^{\mathrm{bi}}_{i}\)). Intuitively, if \(t^{\mathrm{mo}}_{i}\) is large, then \(t^{\mathrm{bi}}_{i}\) would be large too, as the example from Fig. 1 seems to suggest. We thus aim to prove the following.

Theorem 2.2

(Main Inequality)

For any bichromatic arrangement of n pseudo-lines in2,
$$t^{\mathrm{mo}}_i \le\bigl(c_1i+O(1)\bigr) \bigl( \Delta t^{\mathrm{mo}}_i+ \Delta t^{\mathrm{bi}}_i \bigr) + \bigl(c_2+O(1/i)\bigr)t^{\mathrm{bi}}_i+ O(n_ii), $$
where n i denotes the number of curves that have level in [ki,k+i] at x=∞ or x=−∞, and c 1 and c 2 are specific constants with c 2<2.

In our 3-d application, interior bichromatic vertices are fortunately rarer than interior monochromatic vertices, by a factor of 2. We claim that the above inequality implies a subcubic bound for a level in 3-d:

Corollary 2.3

For any arrangement of n pseudo-planes in3, the number of vertices at level k is \(O(n^{3-1/c_{0}})\), where \(c_{0}=\frac{3c_{1}}{2-c_{2}}\), and c 1 and c 2 are the constants in the main inequality.

Proof

We apply Theorem 2.2 to the subarrangement inside each given surface, with the color scheme described above. Observe that the level of each vertex v in the 2-d (bichromatic) subarrangement is equal to the level of v in the 3-d arrangement. Each vertex v lies in three subarrangements. We claim that it is monochromatic in two of them and bichromatic in one of them: Consider the lower envelope of the three surfaces σ 1,σ 2,σ 3 that define v. One edge of the envelope—say, σ 1σ 2—is to the left of v, and the other two are to the right of v, or vice versa. Then v is monochromatic in the subarrangement inside σ 1 and the subarrangement inside σ 2, but bichromatic inside σ 3.

Therefore, summing the left- and right-hand sides of the main inequality over all the n subarrangements gives the following upper bound: So, t i ≤(c 0 i+O(1))(t i+1t i )+O(n 2 i), i.e., \(t_{i} \le(1+\frac{1}{c_{0}i+O(1)})\,t_{i+1} + O(n^{2})\). As base case, t n =O(n 3). Solving this recurrence is straightforward (as described in the Appendix of [8]) and yields \(t_{i}=O(n^{3-1/c_{0}}i^{1/c_{0}})\). □

In Sect. 3, we will establish the 2-d main inequality for a certain choice of constants c 1 and c 2.

2.3 Generalization to Other Surface Families

The preceding proof plan can be adapted to handle other types of surface in 3-d besides pseudo-planes. The precise requirement for the surfaces is as follows:
(∗)

The n given surfaces are graphs of total bivariate functions such that the 2-d subarrangement within each surface (when projected to xy-plane) forms a collection of O(n) x-monotone curve segments in a family with subquadratic cutting number.

Here, a family of curve segments has subquadratic cutting number if any N curve segments in the family can be cut into O(N 2−κ ) pseudo-segments (where each pair intersects at most once), for some constant κ>0.

The restriction to graphs of total functions is not crucial, since we can add near-vertical extensions to make them total.

The cutting number in 2-d arrangements was introduced by Tamaki and Tokuyama [27], who proved the first nontrivial results for pseudo-parabolas (graphs of total univariate functions that pairwise intersect twice); the current best bound for pseudo-parabolas has κ≈1/2 [2, 19]. Other curve families known to have subquadratic cutting number include pseudo-parabolic segments (x-monotone curve segments that pairwise intersect at most twice) with κ=1/3 [7], and graphs of univariate degree-s polynomial functions with κ≈1/2 s−1 [7, 19].

Some examples of surface families that satisfy (∗) thus include the following:
  • pseudo-triangles , which we formally define as a collection of the graphs of n bivariate functions such that after near-vertical extensions are added, each 2-d subarrangement forms a family of O(n) pseudo-segments;

  • pseudo-spherical patches , which we formally define as a collection of the graphs of n bivariate functions such that after extensions are added, each 2-d subarrangement forms a family of O(n) pseudo-parabolic segments;

  • surfaces each having equation z=p(x)+ay for some degree-s polynomial p(x) and constant a.

(In these three examples, we have κ=1, κ≈1/2, and κ≈1/2 s−1, respectively.)

We can apply the same reduction to 2-d (in the proof of Corollary 2.3, we should consider the lower envelope of σ 1,σ 2,σ 3 only locally around v). All that is needed is a main inequality for 2-d curve families with subquadratic cutting number. It is not obvious how to obtain such an inequality for general curves directly without blowing up the coefficient c 2 beyond 2. One idea is to cut the curves first (which increases the number of endpoints) and then prove the same main inequality for pseudo-segments (with n i redefined as the number of pseudo-segment endpoints with level in [ki,k+i]). The overhead term increases (to O(n 3−κ i)), but this can still yield a subcubic result. However, for better results, we follow instead an idea from the previous paper [8] of not explicitly cutting the curves but charging features to certain “lenses”:

Definition 2.4

If two curves γ 1 and γ 2 intersect more than once, the part of γ 1γ 2 between two consecutive intersection points is called a lens. We say that a lens is i-light if every vertical line segment inside the lens intersects at most i red curves and at most i blue curves.

We will prove the following generalization of the main inequality for arbitrary 2-d x-monotone curves.

Theorem 2.5

(Main Inequality)

For any bichromatic arrangement of n curves that are graphs of total univariate functions,
$$t^{\mathrm{mo}}_i \le\bigl(c_1i+O(1)\bigr) \bigl( \Delta t^{\mathrm{mo}}_i+ \Delta t^{\mathrm{bi}}_i \bigr) + \bigl(c_2+O(1/i)\bigr)t^{\mathrm{bi}}_i+ O\bigl(n_i i + |\varLambda_i|\bigr), $$
where Λ i denotes the collection of (c 3 i)-light lenses, n i denotes the number of curves that have level in [ki,k+i] at x=∞ or x=−∞, and c 1,c 2,c 3 are specific constants with c 2<2.

The same result would then hold for curve segments (with n i replaced by O(n)), since we can add near-vertical (upward if red, downward if blue) extensions at the endpoints to make the functions total—this increases |Λ i | by at most O(ni), as this generates at most O(i) extra lenses in Λ i incident to each endpoint.

The collection of lenses Λ i has depth O(i), i.e., a point can be lie on at most O(i) lenses in Λ i . By a random sampling technique from the previous paper [8, Lemma 4.1] (see also [27]), we know that |Λ i |=O(n 2−κ i κ ). The new recurrence in 3-d becomes
$$t_i \le\bigl(c_0i+O(1)\bigr)\Delta t_i + O\bigl(n^{3-\kappa}i^\kappa\bigr). $$
The solution of this recurrence (e.g., see the Appendix of [8]) leads to the following.

Corollary 2.6

For any arrangement of n surfaces in3 satisfying (∗), the number of vertices at level k is \(O(n^{3-1/c_{0}})\) if κ>1/c 0, and O(n 3−κ ) if κ<1/c 0. Here, \(c_{0}=\frac{3c_{1}}{2-c_{2}}\), where c 1 and c 2 are the constants in the main inequality.

3 Proof of the 2-d Main Inequality

It remains to prove the (generalized) main inequality, Theorem 2.5. This is done in the next three subsections.

3.1 The Basic Charging Scheme

We start with a natural but more involved extension of the author’s charging argument for monochromatic 2-d arrangements [8].

Let c be a positive constant, to be determined later. To avoid special cases, we treat the O(n i ) points with level in [ki,k+i] at x=±∞ as boundary vertices. We also pad the arrangement with (c+2)i+1 extra curves below and (c+2)i+1 extra curves above all the given curves of both colors, without creating any new intersections. We introduce some notation and terminology:

Definition 3.1

  • Call a vertex exceptional if it is a vertex of some lens in Λ i , and ordinary otherwise.

  • For any point v, let δ v be the number such that the level of v is k+δ v .

  • For a point v and a curve γ, let χ v,γ be +1 if v violates γ, and −1 otherwise.

  • Given a vertex u defined by curves γ 0 and γ 1 and a vertex v defined by curves γ 1 and γ 2, we say that v is forward w.r.t. u if γ 2 lies between γ 0 and γ 1 at a vertical line slightly to the right (resp. left) of v, assuming that v is to the right (resp. left) of u. Otherwise, v is backward w.r.t. u. (Figure 2(left) shows an example of a forward vertex v and Fig. 2(right) shows an example of a backward vertex v.)
    Fig. 2

    The basic charging scheme

  • Given curves γ 0 and γ 1 of the same color and given a vertical line , let m(γ 0,γ 1,) be the number of curves of the same color as γ 0,γ 1 that lie between γ 0 and γ 1 at ; let j(γ 0,γ 1,) be the number of curves of the opposite color as γ 0,γ 1 that lie between γ 0 and γ 1 at .

  • We say that (γ 0,γ 1,) is within range if j(γ 0,γ 1,)≤ci and both \(\delta_{\gamma_{0}\cap\ell},\delta_{\gamma_{1}\cap\ell}\in[-i,+i]\). Otherwise, (γ 0,γ 1,) is out of range.

Observation 3.2

Given curves γ 0 and γ 1 of the same color and a vertical line intersecting γ 1 at the point v, if (γ 0,γ 1,) is within range, then

Proof

W.l.o.g., say γ 0 is below v and is red (the other cases are similar). By the definition of levels in a bichromatic arrangement, \(m(\gamma_{0},\gamma_{1},\ell)-j(\gamma_{0},\gamma_{1},\ell)\le\delta_{v} -\delta_{\gamma_{0}\cap\ell}+1\). Since \(\delta_{v},\delta_{\gamma_{0}\cap\ell}\in[-i,+i]\), we have m(γ 0,γ 1,)≤j(γ 0,γ 1,)+δ v +i+1≤(c+1)i+δ v +1≤(c+2)i+1. □

We now describe a scheme of charging interior monochromatic vertices to interior bichromatic vertices and boundary vertices (see Fig. 2).

Definition 3.3

Suppose u is an ordinary interior monochromatic vertex u defined by curves γ 0 and γ 1, and v is a vertex defined by curves γ 1 and γ 2 and lies on the vertical line .
  • For an interior bichromatic vertex v, we say that u sends a charge to v if
    1. (A1)

      (γ 0,γ 1,′) is within range for all vertical lines ′ between u and v, and

       
    2. (A2)

      v is forward w.r.t. u.

       
  • For a boundary (monochromatic or bichromatic) vertex v, we say that u sends cij(γ 0,γ 1,) charges to v if (A1) holds (regardless of whether (A2) holds).

Remarks

Note that if u sends a charge to v, then γ 0 cannot cross γ 1 between u and v, because otherwise u would define a lens that is ((c+2)i+1)-light (by condition (A1), with Observation 3.2) and would thus be exceptional for a sufficiently large c 3>c+2 (see Fig. 3(left)). For a similar reason, we cannot have v receiving a charge from both a vertex u left of v and another vertex u′ right of v (see Fig. 3(right)). Consequently, at most one vertex on γ 0 can send a charge to v.
Fig. 3

If more than one vertex on γ 0 were to send a charge to v, then a lens in Λ i would be formed

Lemma 3.4

  1. (i)

    Each ordinary interior monochromatic vertex u sends at least 2ci charges.

     
  2. (ii)

    Each interior bichromatic vertex v receives at most 4(c+1)i+O(1) charges.

     
  3. (iii)

    Each boundary (monochromatic or bichromatic) vertex v receives at most 2c(c+2)i 2+O(i) charges.

     

Proof

  1. (i)

    Suppose u is defined by γ 0 and γ 1. Imagine moving a vertical sweep line from left to right, starting at u. As passes through a bichromatic forward vertex on γ 0γ 1 (w.r.t. u) while (γ 0,γ 1,) stays within range, the “counter” j(γ 0,γ 1,) increases by 1 and a charge is sent from u to that vertex. (On the other hand, as passes through a bichromatic backward vertex, the counter decreases by 1.) As soon as (γ 0,γ 1,) gets out of range, i.e., j(γ 0,γ 1,) reaches ci or passes through a boundary vertex on γ 0γ 1, terminate the sweep. In the latter case, cij(γ 0,γ 1,) charges are sent from u to the boundary vertex. Thus, at least ci charges are sent from u during this left-to-right sweep. Similarly, at least ci charges are sent during a right-to-left sweep.

     
  2. (ii)

    Suppose v is defined by red curve γ 1 and blue curve γ 2. By Observation 3.2, if v receives a charge from a vertex defined by γ 1 and a red curve γ 0 below v, then m(γ 0,γ 1,)≤(c+1)i+δ v +1. Thus, there are at most (c+1)i+δ v +1 candidates for γ 0 below v, and by a symmetric argument at most (c+1)iδ v +1 candidates for γ 0 above v, yielding a total of at most 2(c+1)i+2 charges received by v from vertices on γ 1. Similarly, there are at most 2(c+1)i+2 charges from vertices on γ 2.

     
  3. (iii)
    Suppose v is defined by γ 1 and γ 2. W.l.o.g., say γ 1 is red. By Observation 3.2, j(γ 0,γ 1,)≥m(γ 0,γ 1,)−iδ v −1 for red curves γ 0 below v. Thus, v receives at most the following number of charges from vertices defined by γ 1 and red curves below v: By a symmetric argument, the number of charges received by v from vertices defined by γ 1 and curves above v is at most \((i-\delta_{v})(ci) + \frac{c^{2}i^{2}}{2}+O(i)\), yielding a total of at most 2ci 2+c 2 i 2+O(i) charges received by v from vertices on‘γ 1. Similarly, there are at most 2ci 2+c 2 i 2+O(1) charges from vertices on γ 2.
     
 □
By the above lemma, the total number of charges is at least \(2ci(t^{\mathrm{mo}}_{i}- O(|\varLambda_{i}|))\) (there are O(|Λ i )) exceptional vertices) and is at most \((4(c+1)i+O(1))t^{\mathrm{bi}}_{i}+ (2c(c+2)i^{2}+O(i))(\Delta t^{\mathrm{mo}}_{i}+\Delta t^{\mathrm{bi}}_{i}+O(n_{i}))\). Dividing by 2ci, we get
$$t^{\mathrm{mo}}_i \le \biggl(2+\frac {2}{c}+O(1/i) \biggr)t^{\mathrm{bi}}_i+ \bigl((c+2)i+O(1)\bigr) \bigl(\Delta t^{\mathrm{mo}}_i+ \Delta t^{\mathrm{bi}}_i\bigr) + O\bigl(n_ii+|\varLambda_i|\bigr). $$
We have thus obtained an inequality of the form stated in Theorem 2.5. There is just one (major!) problem: the coefficient c 2 of the \(t^{\mathrm{bi}}_{i}\) term here is greater than 2, regardless of the choice of c, but in order for Corollaries 2.3 and 2.6 to yield any nontrivial bound for levels in 3-d, we need the coefficient to be strictly less than 2.

3.2 Helpers

Despite the apparent problem, we will not abandon the above charging scheme but will amend it by looking for places for improvement. Specifically, the following kinds of configuration (see Fig. 4(left)) help reduce our bound on the number of charges received.
Fig. 4

(Left) A helper. (Right) An example where there are no helpers at v, but there are helpers at u 1 and u 2

Definition 3.5

Suppose v is a vertex defined by curves γ 1 and γ 2 and lies on the vertical line . Let γ 0 be a curve with the same color as γ 1.
  • For a bichromatic interior vertex v, we say that (v,γ 0) is a helper (at v) if
    1. (B1)

      \(m(\gamma_{0},\gamma_{1},\ell)\le(c+1)i+\chi_{v,\gamma _{0}}\delta_{v}+1\), and

       
    2. (B2)

      no intersection of γ 0 and γ 1 sends a charge to v.

       
  • For a (monochromatic or bichromatic) boundary vertex v, we say that (v,γ 0) forms cij(γ 0,γ 1,) helpers (at v) if the same conditions (B1) and (B2) hold.

By inspecting the proofs of Lemma 3.4(ii) and (iii), we immediately see that

Lemma 3.6

  1. (i)

    Each interior bichromatic vertex v receives at most 4(c+1)i−[the number ofhelpers at v]+O(1) charges.

     
  2. (ii)

    Each boundary (monochromatic or bichromatic) vertex v receives at most 2c(c+2)i 2−[the number of helpers at v]+O(i) charges.

     

Is it always possible to find many helpers in an arrangement? A “canonical” example where there are no helpers at a vertex v is shown in Fig. 4(right), but in this example one can find helpers at nearby vertices. This suggests hope of a positive answer ….

We now classify helpers into a few specific types. The list of definitions below is somewhat elaborate, because of the desire to obtain better constants c 1,c 2 in our proof and to handle general non-pseudoline curves.

Definition 3.7

Suppose u is an ordinary (not necessarily interior) monochromatic vertex defined by curves γ 0 and γ 1, and v is a vertex defined by curves γ 1 and γ 2 and lies on the vertical line .
  • For an interior bichromatic vertex v, we say that (v,γ 0) is a strong helper (from u)  if
    1. (C1)

      (γ 0,γ 1,′) is within range for all vertical lines ′ between u and v, and

       
    2. (C2)

      v is backward w.r.t. u.

       
  • For an interior bichromatic vertex v, we say that (v,γ 0) is a moderate helper (from u)  if
    1. (D1)

      (γ 0,γ 1,) is within range, and

       
    2. (D2)

      (γ 0,γ 1,′) is out of range for some vertical line ′ between u and v, and

       
    3. (D3)

      m(γ 0,γ 1,′),j(γ 0,γ 1,′)≤c 3 i for all vertical lines ′ between u and v.

       
    A moderate helper is further classified as a moderate forward or moderate backward helper depending on whether v is forward or backward w.r.t. u.
  • For an interior bichromatic vertex v, any helper (v,γ 0) that is not strong or moderate are classified as a weak helper.

  • For a boundary vertex v, we say that (v,γ 0) forms cij(γ 0,γ 1,) moderate helpers (from u) if (D1)–(D3) hold. These moderate helpers are classified as moderate forward helpers (regardless of whether v is actually forward or backward w.r.t. u).

Remarks

Note that if (v,γ 0) is a strong or moderate helper from u, then γ 0 cannot cross γ 1 between u and v, because otherwise u would define a lens that is (c 3 i)-light (by condition (C1) or (D3)) and would thus be exceptional for a sufficiently large c 3 (like in Fig. 3(left), with “(c+2)i+1” replaced by “c 3 i”). For a similar reason, (v,γ 0) cannot be a strong or moderate helper from both a vertex u left of v and another vertex u′ right of v (like in Fig. 3(right)). Nor can we have both a strong or moderate helper (v,γ 0) from a vertex u left of v, and v receiving a charge from another vertex u′ right of v.

It can then be checked that each helper (v,γ 0) can indeed be only one of the classified types: strong, moderate forward, moderate backward, and weak; and furthermore, a strong or moderate helper is indeed a helper (because (B1) is implied by (C1) or (D1), and (B2) is implied by (C2) or (D2)).

The following lemma is useful in further improving constants. It turns out that strong helpers can not only reduce the number of charges received but also boost the number of charges sent. Furthermore, an abundance of moderate backward helpers automatically imply an abundance of moderate forward helpers.

Lemma 3.8

  1. (i)

    Each ordinary interior monochromatic vertex u sends at least 2ci+[the numberof strong helpers from u] charges.

     
  2. (ii)

    From each ordinary (not necessarily interior) monochromatic vertex u, the number of moderate forward helpers is at least the number of moderate backward helpers.

     

Proof

  1. (i)

    This follows by inspecting the proof of Lemma 3.4(i). During the sweep for vertex u, when passes through a bichromatic backward vertex v on γ 0γ 1, we get a strong helper at v from u and the counter j(γ 0,γ 1,) decreases by 1, allowing for one more subsequent increment (and thus a charge to one more interior vertex) or a lower final value for j(γ 0,γ 1,) (and thus an extra charge to a boundary vertex).

     
  2. (ii)

    Perform the same left-to-right sweep starting at u, except this time the sweep is terminated only when j(γ 0,γ 1,) or m(γ 0,γ 1,) exceeds c 3 i. The regions swept by during which (γ 0,γ 1,) is within range form a union of disjoint “windows”. Take one such window, excluding the initial window containing u (there is no initial window if u is not an interior vertex). When passes through a bichromatic forward (resp. backward) vertex v on γ 0γ 1 within this window, we get a moderate forward (resp. backward) helper at v from u and the counter j(γ 0,γ 1,) increases (resp. decreases) by 1. At the right boundary of the window, j(γ 0,γ 1,) reaches ci or passes through a boundary vertex v on γ 0γ 1. In the latter case, we get cij(γ 0,γ 1,) moderate forward helper at v. Thus, there are at least as many moderate forward helpers as moderate backward helpers within each window. A similar argument holds for a right-to-left sweep.

     
 □

Let H strong, H for, H back, and H weak denote the number of strong, moderate forward, moderate backward, and weak helpers, respectively.

By Lemmas 3.8(i) and 3.6, the total number of charges is at least \(2ci(t^{\mathrm{mo}}_{i}- O(|\varLambda_{i}|)) + H_{\mathrm{strong}}\) and is at most \((4(c+1)i+O(1))t^{\mathrm{bi}}_{i}+ (2c(c+2)i^{2}+O(i))(\Delta t^{\mathrm{mo}}_{i}+\Delta t^{\mathrm{bi}}_{i}+O(n_{i})) - H_{\mathrm{strong}} - H_{\mathrm{for}} - H_{\mathrm{back}} - H_{\mathrm{weak}}\). Dividing by 2ci and using the fact that H forH back≥0 by Lemma 3.8(ii), we get

3.3 A More Sophisticated Charging Scheme

We now prove the abundance of helpers by devising a second charging scheme, this time, with charges sent from interior bichromatic vertices to helpers. Let α>2/c and β∈(α,1−α) be constants, to be set later.

Sweeping

Take an ordinary interior bichromatic vertex v, defined by red curve γ 1 and blue curve γ 2. W.l.o.g., say γ 1 is below γ 2 slightly to the left of v (the other case is symmetric). Let h be the number of helpers of the form (v,γ) with γ red and below v, or blue and above v.

Move a vertical sweep line from right to left, starting at v. Maintain the following counters (see Fig. 5(left)):
  • Let j 1 (resp. j 2) be the number of blue (resp. red) curves between γ 1 and γ 2.

  • Let m σ (resp. \(m'_{\sigma}\)) be the number of bichromatic forward (resp. backward) vertices on γ σ between and v (σ∈{1,2}). Let m=m 1+m 2.

  • Let p σ (resp. \(p_{\sigma}'\)) be the number of monochromatic forward (resp. backward) vertices on γ σ between and v.

As passes through a bichromatic forward vertex on γ 1γ 2, the counter m increases by 1. As soon as hits a boundary vertex on γ 1γ 2 or m reaches (1−α)cih−1, terminate the sweep.
Fig. 5

The more sophisticated charging scheme. (Left) Counters \(j_{\sigma},m_{\sigma},m_{\sigma}',p_{\sigma},p_{\sigma}'\). (Right) Case 1: a strong or moderate backward helper (w,γ 2) from u

Observation 3.9

\(j_{1}+m_{1}',\,j_{2}+m_{2}'\le m+i+\delta_{v}+1 < ci\).

Proof

By the definition of levels, \(m_{2}+p_{2}'-m_{2}'-p_{2}\ge\delta_{\gamma_{2}\cap\ell}-\delta_{v}-1\). Thus, \(j_{1}=m_{1}+p_{2}-m_{1}'-p_{2}'\le m_{1}+m_{2}-m_{1}'-m_{2}'+\delta_{v}-\delta_{\gamma _{2}\cap \ell}+1\). Since \(\delta_{v},\delta_{\gamma_{2}\cap\ell}\in[-i,+i]\), we have \(j_{1}+m_{1}'\le m+i+\delta_{v}+1\le(1-\alpha)ci + 2i < ci\). The other inequality for \(j_{2}+m_{2}'\) is similar. □

Remark

Note that during the right-to-left sweep, γ 1 cannot cross γ 2, because otherwise v would define a lens that is (ci)-light (by Observation 3.9) and would thus be exceptional.

Actually, in the above sweep, cannot hit a boundary vertex on γ 1γ 2. To see this, suppose reaches a boundary vertex w on γ 1 (the other case is similar). For at least (c+1)i+δ v +1−h red curves γ below v, some intersection y γ of γ and γ 1 sends a charge to v (by the definition of helpers and the number h); and y γ must be to the right of w (in order for y γ to send a charge to v). Now, γ must cross the vertical line segment between w and γ 2, or cross γ 2 between γ 2 and v, because otherwise γ would cross γ 1 between w and y γ , and y γ would define a lens that is (ci)-light (since j 1 and j 2 always stay below ci by Observation 3.9) and would thus be exceptional. It follows that \(j_{2}+m_{2}'\) reaches at least (c+1)i+δ v +1−h, and by Observation 3.9, m reaches at least cih>(1−α)cih−1: a contradiction.

Charging

For each of the h helpers of the form (v,γ) with γ red and below v, or blue and above v, as initialization, v sends 1 unit of charge to the helper.

Suppose passes through a forward interior bichromatic vertex w on γ 1γ 2. W.l.o.g., say w is defined by γ 1 and a blue curve γ 0 (the other possibility is similar). The heart of the proof lies in the following case analysis (see Figs. 5 and 6).
Fig. 6

The more sophisticated charging scheme, continued. (LeftCase 2: a strong, moderate backward, or weak helper (v,γ 0). (RightCase 3: strong helpers (z γ ,γ 1) from y γ for many γ’s

Lemma 3.10

At least one of the following cases must hold:
  • Case 0: w is an exceptional vertex.

  • Case 1: (w,γ 2) is a strong or moderate backward helper.

  • Case 2: (v,γ 0) is a strong, moderate backward, or weak helper.

  • Case 3: At some vertical line between w and v, we have γ 0 below γ 1, and there are more than ci blue curves or more than (c+2)i+1 red curves between γ 0 and γ 1.

Proof

Suppose that Case 3 does not hold. Then at every vertical line ′ between w and v, if γ 0 is below γ 1, there are at most ci blue curves and at most (c+2)i+1 red curves between γ 0 and γ 1. In particular, if γ 0 is below γ 1 at , there are at most ci≤(c+1)i+δ v +1 blue curves between γ 0 and γ 1 at .

We may assume that γ 0 and γ 1 do not cross between w and v (and thus γ 0 is indeed below γ 1 at ), because otherwise w would define a lens that is ((c+2)i+1)-light and we would be in Case 0.

If v receives a charge from a vertex u on γ 0, or if (v,γ 0) is a moderate forward helper from u, then (w,γ 2) is a strong or moderate backward helper from u and we are in Case 1 ((D1) and (D3) hold since j 1 and j 2 stay below ci by Observation 3.9, as one can see from Fig. 5(right) for a sufficiently large c 3>2c+2). On the other hand, if (v,γ 0) is a strong, moderate backward, or weak helper, then we are in Case 2. □

Our charging scheme is as follows:
  • In Case 0, v sends 1 unit of charge to the vertex w itself.

  • In Case 1, v sends 1 unit of charge to the helper (w,γ 2).

  • In Case 2, v sends 1 unit of charge to the helper (v,γ 0).

  • In Case 3, note that for at least (c+1)i+δ v +1−h red curves γ below v, some intersection y γ of γ and γ 1 sends a charge to v (by definition of helpers and the number h). If y γ is to the right of w, then γ must cross the vertical line segment between w and γ 2, or cross γ 2 between γ 2 and v, because otherwise γ would cross γ 1 between w and y γ , and y γ would define a lens that is (ci)-light (since j 1 and j 2 always stay below ci by Observation 3.9) and would thus be exceptional. Thus, by excluding at most \(j_{2}+m_{2}'\) candidates for γ, we can ensure that y γ is to the left of w.

    Since y γ sends a charge to v, there are at most ci blue curves and at most (c+2)i+1 red curves between γ and γ 1 at ′ (by (A1), with Observation 3.2). In particular, γ must be above γ 0 at ′, due to the condition stated in Case 3. Thus, some intersection z γ of γ and γ 0 must be backward w.r.t. y γ , and so (z γ ,γ 1) is a strong helper from y γ (see Fig. 6(right); condition (C1) holds for this helper because y γ sends a charge to v). The number of candidates for γ is at least \((c+1)i+\delta_{v}+1-h-j_{2}-m_{2}'\ge ci-h-m\) by Observation 3.9. We make v send \(\frac{1}{\beta ci}\) units of charge to each such strong helper.

This completes the description of the right-to-left sweeping and charging process for v. We perform a similar process for v, this time, sweeping from left to right.

Analysis

Each ordinary interior bichromatic vertex v sends at least the following number of charges in the right-to-left sweep (including the h initial charges): It can be checked that this last expression is monotone increasing in h (ignoring the O(1) terms), and so the number of charges is lower-bounded by the value in the first case, i.e., \((1-\frac{\beta}{2} - \frac{\alpha^{2}}{2\beta} )ci - O(1)\). Therefore, v sends a total of at least (2−βα 2/β)ciO(1) charges in both sweeps.
In regard to the number of charges received:
  • Each exceptional vertex w receives O(i) charges due to Case 0, since m 1,m 2≤(1−α)ci implies that given w, there are O(i) candidates for γ 2, and thus for v.

  • Each strong helper receives 1 charge initially, at most 1 charge due to Case 1, at most 1 charge due to Case 2, and at most \((1-\alpha)ci\frac{1}{\beta ci} = (1-\alpha)/\beta\) charges due to Case 3, since m 1,m 2≤(1−α)ci implies that given (z γ ,γ 1), there are at most (1−α)ci candidates for γ 2, and thus for v.

  • Each moderate forward helper receives 1 charge during initialization and no more afterwards.

  • Each moderate backward helper receives 1 charge during initialization, at most 1 charge due to Case 1, and at most 1 charge due to Case 2.

  • Each weak helper receives 1 charge during initialization, and at most 1 charge due to Case 2.

To summarize, we have shown that the total number of charges in the above scheme is at least \(((2-\beta-\alpha^{2}/\beta) ci-O(1))(t^{\mathrm{bi}}_{i} -O(|\varLambda_{i}|))\) and is at most \((3\hspace{-0.5pt}+\hspace{-0.5pt}(1\hspace{-0.5pt}-\hspace{-0.5pt}\alpha)/\beta)H_{\mathrm{strong}}\hspace{-0.5pt}+\hspace{-0.5pt}H_{\mathrm {for}}\hspace{-0.5pt}+\hspace{-0.5pt}3H_{\mathrm{back}}\hspace{-0.5pt}+\hspace{-0.5pt}2H_{\mathrm{weak}}\hspace{-0.5pt}+\hspace{-0.5pt}O(i|\varLambda_{i}|)\). Since 3+(1−α)/β>4, The coefficient \(\frac{2-\beta-\alpha^{2}/\beta}{3+(1-\alpha)/\beta}\) exceeds 0.310102 by setting α=0.15505 and β=0.53485. (Note that indeed β∈(α,1−α).)
So finally, inequality (2) with λ=0.25 yields

The coefficient of the \(t^{\mathrm{bi}}_{i}\) term here is strictly less than 2 for a sufficiently large choice of the parameter c. The proof of Theorem 2.5 is now complete. For Corollaries 2.3 and 2.6, we have \(c_{0}=\frac{3(c+2)}{0.310102-2/c}\), which is below 286.97 by setting c=13.8312, with c 1=15.8312 and c 2<1.834499. (Note that indeed α>2/c.)

4 Final Remarks

Just like in earlier work on levels in arrangements of curves, we have shown how a nontrivial cutting number bound translates to a nontrivial level bound for any family of surfaces in 3-d. In light of this result, we reiterate the following open problem [7]: do general fixed-degree algebraic curves in the plane have a subquadratic cutting number? If so, a subcubic level bound would immediately follow for graphs of fixed-degree bivariate polynomial functions in 3-d.

Perhaps it might be possible to obtain slight improvements on the constants in the 2-d main inequality by lengthening the proof with an even more detailed case analysis, but it would be more desirable to find a simpler yet smarter charging argument that could yield more drastic improvements. An intriguing question is to determine what is the smallest value c 2 attainable in the 2-d inequality. Alternatively, can one prove the 3-d inequality directly?

It is doubtful that our approach could improve known upper bounds for levels for planes (i.e., the k-set problem) in 3-d. A more intriguing direction to pursue would be the case of hyperplanes in higher dimensions, where the previous upper bounds are very weak. Unfortunately, our reduction to 2-d fails to yield o(n d ) bounds as soon as the dimension d reaches 4, because there could be as many bichromatic vertices (\(t^{\mathrm {bi}}_{i}\)) as monochromatic vertices (\(t^{\mathrm{mo}}_{i}\)) on average in the resulting 2-d subarrangements, and so we need the coefficient c 2 to be strictly less than 1—an impossible demand (as the example in Fig. 1 indicates). Still, it is possible to adapt the approach of this paper to obtain new k-sensitive upper bounds for hyperplanes in 4-d for a certain range of k values, as the author has shown in a recent paper [11]. New k-sensitive upper bounds might also be possible in higher dimensions if we could somehow get c 2 closer to 1. In 2-d and 3-d, our approach has been shown [11] to lead to new results for a bichromatic version of the k-set problem.

Footnotes

  1. 1.

    In fact, the bound presented here is already an improvement over the O(n 3−1/705.48)=O(n 2.9986) bound that was originally announced in the conference version of this paper [9].

Notes

Acknowledgement

This work was supported in part by an NSERC grant.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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