On Conflict-Free Coloring of Points and Simple Regions in
∗
the Plane
Sariel Har-Peled
Shakhar Smorodinsky
Department of Computer Science, DCL 2111
University of Illinois
1304 West Springfield Ave.
Urbana, IL 61801; USA.
School of Computer Science,
Tel-Aviv University.
Tel-Aviv, Israel
[email protected]
[email protected]
ABSTRACT
General Terms
In this paper, we study coloring problems related to frequency assignment problems in cellular networks. In abstract setting, the problems are of the following two types:
Algorithms, Theory.
Keywords
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 conflictfree coloring of S (CF-coloring in short).
Cellular Network, Coloring, Frequency Assignment.
1. INTRODUCTION
The study of the problems mentioned above, which originated in [10] and [17], was motivated by the problem of
frequency assignment in cellular networks. 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 network is to assign
frequencies to base-stations so that every client is served by
some base-station, in the sense that it lies within the range
of the station and no other station within its reception range
has the same frequency. The goal is to minimize the number of assigned frequencies since the frequency spectrum is
limited and costly.
Suppose we are given a set of n antennas. The location of
each antenna (base station) and its radius of transmission is
fixed and is given (as a disc in the plane). Even et al. [10]
have shown that one can find an assignment of frequencies
to the antennas with a total of at most O(log n) frequencies
such that each antenna (a base station) is assigned one of the
frequencies and the resulting assignment is free of conflicts,
in the sense that, for each point p in the plane, there exists
at least one antenna a, whose disc of transmission contains
p so that a’s frequency is different from those of all other
antennas that can reach p. Furthermore, it was shown that
this bound is worst-case optimal.
Thus, Even et al. have shown 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. The approach
used in [10] relies strongly on the fact that the regions under
consideration are discs.
In this paper, we improve and extend results of [10] com-
CF-coloring of a range space: Given a set P of n
points in Rd and a set R of ranges (for example, the set
of all discs in the plane), what is the minimum integer
k, such that one can color the points of P by k colors,
so that for any r ∈ R with P ∩ r = ∅, 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).
Categories and Subject Descriptors
I.3.5 [Computational Geometry]: Computational Geometry and Object Modeling
∗The full version of the paper is available from [11]. Work
on this paper was done while the second author was a Ph.D
student under the supervision of Prof. Micha Sharir.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
SoCG’03, June 8–10, 2003, San Diego, California, USA.
Copyright 2003 ACM 1-58113-663-3/03/0006 ...$5.00.
114
note that, without the assumption that the rectangles are
axis-parallel, the problem becomes uninteresting. Indeed,
any planar set P of n points in general position (i.e., no
three are co-linear) needs n colors in any CF-coloring of P
with respect to arbitrarily oriented rectangles. Indeed, in
such a set, any pair of points p, q ∈ P need to be colored
with distinct colors since there is a rectangle (a slight expansion of the segment pq) containing only p and q and no
other point of P . We also study the case of random point
sets inside a square.
bining more involved probabilistic and geometric ideas. Our
main result, which is delegated to Section 4 is a general
probabilistic algorithm which CF-colors any set of n “simple” regions (not necessarily convex) whose union has “low”
complexity, using a “small” number of colors. In particular, we show that if the regions under consideration have
a union of near linear complexity, then they can be CFcolored using polylogarithmic number of colors. This holds
for pseudo-disks [13], convex α-fat [8] shapes, and (α, β)covered objects [7]. This provides the first non-trivial and
near-optimal bounds for one of the problems that motivated
the work of Even et al.. In practice, since cellular antennas
are in fact directed, the region of influence of an antenna is a
circular sector with central angle of 60o . Since such sectors
are fat and convex, our results thus imply that those regions
have a conflict-free coloring using a polylogarithmic number
of colors.
In Section 4.1, we further refine those results, deriving
better bounds for some special cases. We show that any set
of n axis-parallel rectangles in the plane can be CF-colored
with O(log2 n) colors. Thus, improving and generalizing a
result of Even et al. [10], where only the case of rectangles with bounded size ratio was studied. We note (as in a
similar observation made above) that the assumption that
the rectangles be axis-parallel cannot be removed, for otherwise one can construct a set R of n rectangles in which
any CF-coloring of R needs n colors. This is done by placing n straight line segments such that each pair intersect in
a distinct point. Then, replacing each such segment by a
“narrow” rectangle, we are left with a set R of n pairwise
intersecting rectangles such that each pair has a point in
common only to that pair. This implies that in any CFcoloring of R, any two rectangles need to be colored with
distinct colors and the claim follows.
In Section 5, we generalize the notion of CF-coloring of
range spaces to what we call k-CF-coloring. That is, we say
that a range is “served” if there is a color that appears in the
range at most k times, for some fix pre-specified parameter
k. This notion seems to be natural and provides additional
flexibility. A similar generalization of CF-coloring a set of
regions is also studied. For example, we show that there is a
range space consisting of n points for which any CF-coloring
√
needs n colors but there exists a 2-CF-coloring with O( n)
1/k
colors (and a k-CF-coloring with O(n ) colors for any fixed
k ≥ 2).
In Section 3 we study the problem of CF-coloring of range
spaces, where the underlying ranges are axis-parallel rectangles in the plane,√ and show that any n points can be
CF-colored with O( n) colors with respect to axis-parallel
rectangles. (Note the sharp contrast with the “dual” problem of CF-coloring rectangles, where a bound of O(log2 n)
is known.) Using a different approach, we also obtain nontrivial upper bounds on the number of colors needed in any
CF-coloring of a range space consisting of n points in Rd
whose ranges are axis-parallel boxes. We also study√the √
special case when all the given points form the regular n× ngrid and show that in this case one can color the points
with O(log n) colors and that this bound is worst-case optimal. This bound holds for any dimension. Namely for any
fixed d one can color the points of the d-dimensional regular
n1/d × . . . × n1/d grid with O(log n) colors with respect to
axis-parallel boxes. In fact, we show that the constant in the
big ‘O’ notation does not depend on the dimension d. We
2. PRELIMINARIES
We briefly introduce some notations and tools used in this
paper. In the following, P denotes a set of n points in Rd ,
and R denotes a set of ranges (for example, the set of all
discs in the plane). We refer to the pair (P, R) as a range
space. The “Delaunay” graph G = G(P, R) is the graph
whose vertex set is P and whose edges are all pairs (u, v) for
which there exists a range r ∈ R, such that {u, v} = r ∩ P .
We denote a range realizing an edge (u, v) ∈ G by ruv .
A coloring f : P → {1, . . . , k} is a conflict-free coloring of
(P, R) (CF-coloring in short), if for any r ∈ R, such that
P ∩ r = ∅, there exists a color i, for which there is a point
p ∈ P ∩ r, such that f (p) = i, and no other point of P ∩ r
is assigned the color i. Any range r for which this property
holds (regardless of whether the coloring is conflict free) is
said to be served by the coloring. We refer to the minimum
number of colors needed to CF-color (P, R) as the conflictfree (or CF)-chromatic number of (P, R).
For a set R of ranges in Rd let kopt (n, R) denote the maximum number of colors needed for the given set R, over all
sets of n points in Rd .
Even et al. have shown that the problem of CF-coloring
a family S of n discs in the plane can be reduced to that of
CF-coloring a range space (P, R) where P is a set of n points
in R3 and R is the set of all half-spaces. Thus, CF-coloring
of geometric range-spaces are of independent interest.
Definition 2.1 A range space (P, R) is called monotone if
for each r ∈ R with |r ∩ P | > 2 there exists a range r ∈ R
such that:
1. |r ∩ P | = 2,
2. r ∩ P ⊂ r ∩ P .
It is easy to verify that this property holds when R is the
set of all axis-parallel rectangles in the plane.1
A natural approach (used in [10]) for conflict-free coloring
of (P, R), is to pick a large independent set L1 in G(P, R),
color all the points of L1 by a single color, and repeat this
process on (P \ L1 , R). We summarize this approach in
Algorithm 1.
Let Li ⊂ P denote the set of points in P colored with i
by Algorithm 1. We refer to Li as the i-th layer of (P, R).
Lemma 2.2 The coloring of a monotone range space (P, R)
by Algorithm 1 is a valid CF-coloring of (P, R).
Proof. Consider a range r ∈ R, such that |P ∩ r| ≥ 2.
Let i be the maximal color of the points in r. Let Pi ⊂ P be
1
The interested reader might try to prove this property for
the case where R is the set of all discs in the plane.
115
Algorithm 1 CFcolor(P, R) - CF-color a set P with respect
to a set of ranges R.
1: i ← 0. (i denotes an unused color)
2: while P = ∅ do
3:
Find an independent set P ⊂ P of G(P, R). (We
elaborate subsequently on the implementation of this
step.)
4:
Color: ∀x ∈ P : f (x) ← i.
5:
Remove: P ← P \ P 6:
Increment: i ← i + 1.
7: end while
Lemma 2.5 (i) Let G be a simple graph on n vertices
with average degree δ. Then G contains an independent set of size Ω(n/δ).
(ii) Let H be a k-uniform hypergraph with n vertices and
average degree
δ. Then
H contains an independent set
of size Ω n/δ 1/(k−1) .
Proof. Both facts are easy exercises in graph theory
(see, e.g., [2]); nevertheless we provide proofs for the sake
of completeness. (Note also that part (i) is a special case of
part (ii), with k = 2.)
(i) Remove from G all vertices of degree larger than, say,
2δ. Let G be the resulting graph. Clearly, the number
of vertices in G is at least n/2, and the maximum degree in G is at most 2δ. Thus, G contains an independent set of size Ω(n/δ), which is also an independent
set in G.
the set of input points at the beginning of the ith iteration,
i.e., the set just before color i has been assigned. Note that
Li ⊂ Pi and Li ∩ r = Pi ∩ r (since i is the maximal color in
r). Clearly, if |r ∩ Li | = 1 then r is served and we are done.
Thus, we only have to consider the case |r ∩ Li | > 1.
However, by the monotonicity property (applied to the set
Pi ), it follows that there exists a range r such that: (i)
|r ∩ Pi | = |r ∩ Li | = 2, and (ii) r ∩ Pi ⊂ r ∩ Pi = r ∩ Li .
This means that the two points of r ∩ Li form an edge in
the graph G(Pi , R). This however contradicts the fact that
Li is independent in G(Pi , R), and thereby completes the
proof of the lemma.
(ii) Let H = (V, E) be the given k-uniform hypergraph.
Let m = |E| be the number of edges of H. By direct
counting, we have m = δn/k.
Let S be the set of vertices generated by picking each
vertex of V independently with probability p, where
p will be specified shortly. For each edge e ∈ E that
is fully contained in S , we remove one of its vertices
from S . Let S denote the resulting set, which is clearly
independent. We have E[|S|] ≥ pn−pk m = pn−pk δn
.
k
Setting p = (k/2δ)1/(k−1) , we have
To realize the usefulness of Lemma 2.2, consider the following result of [10]: Let R be the set of all discs in the
plane. Then, the chromatic number of (P, R) is O(log n),
where n = |P |. The proof follows immediately from the fact
that (P, R) is monotone and Lemma 2.2, as G(P, R) is just
the Delaunay graph of P , which is planar (see e.g., [6]), and
as such it has an independent set of size at least n/4 (by the
four colors theorem). It follows, that P has a decomposition into O(log n) layers and hence the chromatic number of
(P, R) is O(log n). Pach and Tóth [15] have recently shown
that any set P of n points in the plane, needs Ω(log n) colors
in any CF-coloring of (P, R).
We summarize this technique in the following lemma.
E[|S|]
≥ n
= n
k
2δ
k
2δ
1/(k−1)
−
1/(k−1) δn
k
1−
1
2
k
2δ
k/(k−1)
.
It follows, by the probabilistic method,
that there ex
ists an independent set in H of size Ω n(k/δ)1/(k−1) =
Ω n/δ 1/(k−1) .
Lemma 2.3 Let R be a set of ranges in Rd . If R has the
property that any finite range space (P, R) is monotone and
the Delaunay graph G(P, R) contains:
(i) an independent set of size at least α|P |, for some fixed
log n
.
0 < α < 1, then kopt (n, R) ≤ log(1/(1−α))
(ii) an independent set of size Ω(|P | ), for some fixed 0 <
< 1, then kopt (n, R) = O(n1− ).
.
3. CF-COLORING OF POINTS WITH RESPECT TO RANGES
3.1 Axis Parallel Rectangles
Proof. The assumption in part (i) of the lemma implies
that in the ith iteration of Algorithm 1 we color at least
α|Pi | points of Pi with the color i. This means that if we
start with a set of n points, the number of iterations is at
log n
. Similarly, in part (ii) of the lemma we
most log(1/(1−α))
observe that the number of iterations needed by Algorithm 1
is bounded by O(n1− ).
In this section, we deal with the problem of conflict-free
coloring of points in the plane, where the ranges are axisparallel rectangles.
Theorem 3.1 For the set R of all axis-parallel
rectangles
√
in the plane, we have kopt (n, R) = O( n).
Proof. Let P be a set of n points in the plane, and
let G = G(P, R) denote the corresponding Delaunay
graph.
√
If there is a point p ∈ P with degree ≥ 2 n in G, then
there are two opposite
quadrants around p that contain
√
together at least n neighbors of p in G(P, R). See Figure 1. Suppose, without loss of generality that these are
the upper-right and the lower-left quadrants. The neighbors of p in each of the quadrants form a monotone decreasing sequence. Choosing every other element in each
Definition 2.4 For a finite set V , a k-uniform hypergraph
H on V consists of V together with a set E of subsets of V
of size k (those are the hyper-edges of H). The degree of a
vertex v ∈ V is the number of sets (i.e., hyper-edges) of E
that contain v.
A set A ⊆ V is called an independent set if no hyper-edge
of E is contained in A. See [3] for more details concerning
uniform hypergraphs.
116
In contrast to the rather weak bounds of Theorem 3.1 and
Theorem 3.3, we next show that the special case where P is
a grid admits a CF-coloring of (optimal) logarithmic size.
p
Definition 3.4 Let Bd denote the set of all axis-parallel
boxes in Rd . The grid G(n, d) is the Cartesian product
n
j
1, . . . , n1/d
sequence
yields an independent set in G of size at least
√
√
n/2. Otherwise, all the points of p have degree < 2 n
in G. However, in this case, Lemma√2.5 implies that there
is an independent set in G of size Ω( n). Thus, the number
T (n) of layers into√which P can be decomposed satisfies:
for an appropriate constant c. √
This
T (n) = 1 + T (n − c n), √
implies that T (n) = O( n) + T (n/2), or T (n) =√O( n).
By Lemma 2.2, (P, R) can be CF-colored using O( n) colors.
Proof. We only prove the lower bound. The other claims
can be easily verified. Let f (·) be any conflict-free coloring of
G(n, 1) = {1, . . . , n}, using the minimum number of colors.
Let h(m) be the minimum number of colors used by f (·) for
coloring an interval of length m.
Consider the color appearing exactly once in the coloring
f (·) of the interval I = [1, n]. Namely, there is an i ∈ I such
that f (i) = f (j), for all j ∈ I, j = i. Let Il = [1, i − 1] and
Ir = [i + 1, n], and assume, without loss of generality, that
|Ir | ≥ |Il |. Clearly, we have h(n) = h (|I|) ≥ 1 + h (|Ir |) =
h ((n − 1)/2) + 1. By induction, it is now easy to prove
that h(n) ≥ log n + 1.
Remark 3.2 Noga Alon, Timothy Chan, János Pach and
Geza Tóth (personal communications) have independently
noticed that the result of Theorem 3.1 can be slightly improved by a polylogarithmic factor, using more involved
graph-theoretic arguments [1, 15].
Lemma 3.6 The conflict-free chromatic number of (G(n, d),
Bd ) is at most 1 + log n.
Substantially improving the result of Theorem 3.1 is the
main open problem that we pose in this paper, as we currently have only a trivial lower bound of Ω(log n).
Using a somewhat different approach, we next give an
alternative proof of Theorem 3.1. This later approach generalizes to higher dimensions:
Proof. For i = (i1 , . . . , id ) ∈ G(n, d), we define its color
P
to be f (i) = dj=1 cf(ij )−(d−1). Let R be any axis-parallel
box, and let Nj be the set of integers in the projection of R
into the j-th axis. Note that, for j = 1, . . . , d, cf(Nj ) has
a unique maximum on this range, by Lemma 3.5. Let ij
be the index that realizes it. Clearly, f (i ) is the maximum
value achieved by f (·) on R, where i = (i1 , i2 , . . . , id ). Furthermore, no other point of R ∩ G(n, d) realizes this value.
Thus f (·) provides the required conflict-free coloring. To
complete the proof, jnote that kthe value of f (·) is bounded
from above by d(1 + log(n1/d ) ) − (d − 1) ≤ 1 + log n.
Theorem 3.3 Let R be the
all axis-parallel boxes in
set of
d−1
Then kopt (n, R) = O n1−1/2
.
Rd .
Note that for d = 2 we obtain the same bound as in
Theorem 3.1.
Proof. Let P be a set of n points in Rd , and denote the
coordinates by x1 , . . . , xd . Let P1 be the ordered sequence
of the points of P according to their x1 -coordinate. At the
i-th stage, for i = 2, . . . , d, let Pi be the longest monotone
subsequence of Pi−1 , according to their xi -coordinates. By
the Erdős-Szekeres Theorem (See, e.g., [18])
a
pthere exists
monotone subsequence of Pi−1 of length Ω
|Pi−1 | .
d−1
Remark 3.7 Lemma 3.6 is tight for d = 1 and d = log n.
For other values of d, one can show a lower bound of log n−
d. To see this, consider any CF-coloring of G(n, d), and let
p be the point with a unique color in the whole grid. Then
there is a box that avoids p and contains almost half of the
points of G(n, d). Analyzing carefully the number of points
remaining in this box, and using induction, we obtain the
aforementioned lower-bound.
points which is
monotone in all coordinates (in each coordinate it can be
either increasing or decreasing). It is easy to verify that if
we pick every other point in this sequence, we obtain an
d−1
.
independent set in G(P, R) of size |Pd |/2 = Ω n1/2
We thus conclude, by Lemma 2.3, that
d−1
kopt (n, R) = O n1−1/2
.
Lemma 3.5 Let I = B1 be the set of intervals on the real
line and let cf(i) be the function defined on the positive integers and returning j + 1 if 2j is the largest power of 2 that
divides i.
Then the conflict-free chromatic number of (G(n, 1), I) is
1 + log n, it is realized by cf(·), and this bound is tight.
Furthermore, for an interval I = [i, j], the color that appears
exactly once in I ∩ G(n, 1) is the largest number in the set
{cf(i), cf(i + 1), . . . , cf(j)}.
Figure 1: A point p and the neighbors of p in two
opposite quadrants in the graph G(P, R).
Thus, Pd is a sequence of Ω n1/2
kod
3.2 Random Point Set Inside a Square
In the following, let U denote the unit square in the plane.
In this section, we consider the conflict-free coloring of a
point-set generated by picking points uniformly and independently out of U.
.
117
of P1 by their given colors, and coloring P2 by additional
O(log 3 n) colors, as specified by the coloring of G2 .
We claim that f (·) is a conflict-free coloring of P1 ∪ P2 .
Indeed, let R ⊆ U be an arbitrary axis-parallel rectangle. If
R ∩ P1 = ∅, then we are done, because the given coloring
of P1 is conflict-free. Furthermore, if area(R) ≥ ε, then it
contains a point of P1 , as P1 is an ε-net.
Thus, it must be that area(R) ≤ ε, R ∩ P1 = ∅, and
R ∩ P2 = ∅. However, by construction of G2 , all the pairs of
points of R ∩ P2 are connected in G2 , thus f (·) assigns all
of them unique colors.
It follows that f (·) is a conflict-free coloring of P1 ∪ P2
with high probability.
p"
p
Theorem 3.9 Let P be a set of n points picked randomly
and uniformly out of the unit square U. Then, with high
probability, the set P have a conflict-free coloring using O(log 4 n)
colors.
Figure 2: The region S(p, ε).
Lemma 3.8 Let P1 , P2 ⊆ U be two random point-sets of
cardinality m each, and assume that P1 was conflict-free
colored using χ colors, where m ≤ n. Then P1 ∪ P2 can
be conflict-free colored using χ + O(log3 n) colors, with high
probability.
Proof. Order P in an arbitrary order, and let Pi be the
first 2i points of P . Now, repeatedly apply Lemma 3.8 to
Pi \ Pi−1 and Pi−1 , for i = 1, . . . , log n.
Observe, that since the property of having an empty axis
parallel rectangle is uniquely defined by the ordering of the
given points in each coordinate. It follows, that instead
of picking points randomly in the unit square, we can just
generate the points by picking a random permutation π of
1, . . . , n, and placing the ith point at (i, π(i)). One can
modify the proof of Theorem 3.9 so that it holds also in this
setting. This results in an identical result with a combinatorial proof instead of a geometric one.
Proof. Clearly, if m = O(log 3 n), then the claim trivially
holds. Otherwise, let ε = O((log n)/m). By ε-net theorem
[12], P1 is an ε-net for rectangles inside the unit square under
the measure of area, with high probability.
For a point p ∈ U, let
n S(p, ε) = q area(rect(p, q)) ≤ ε
o
be the set of points that form rectangles of area at most ε
with p, namely q ∈ S(p, ε) iff |qx − px | · |qy − py | ≤ ε, see
Figure 2. Furthermore,
A = area (S(p, ε)) ≤ 4 ε2 +
= O ε log
1
ε
=O
Z
4. CONFLICT FREE COLORING OF REGIONS WITH LOW UNION COMPLEXITY
1
√
x= ε
ε
dx
x
m
log n
log
m
log n
In this section we present the main result of our paper.
We introduce a general approach which allow us to derive
near optimal bounds on the CF-chromatic number of any
finite collection of regions with “low” union complexity. Our
approach can be applied also to a general geometric rangespace (not necessarily monotone) whose “Delaunay graph”
has “low” complexity.
Let R be a family of regions in the plane, such that the the
complexity of the union of any n regions of R is O(U(n)). In
the following, we assume that U(n) is a near linear function.
This holds for pseudo-disks [13], convex α-fat shapes [8], and
(α, β)-covered objects [7].
.
Clearly, E[|S(p, ε) ∩ P2 |] = Am, and by the Chernoff inequality,
h
Pr |S(p, ε) ∩ P2 | ≥ Am(1 + log n)
≤
≤
i
elog n
(1 + log n)(1+log n)
elog n
(1 + log n)(1+log n)
≤ e−c log
2
n log log n
Am
O(log n)
≤ n−c log n log log n ,
Definition 4.1 For a set S of n regions of R, a subset Sb ⊆
S is admissible (with respect to S), if any p ∈ R2 , satisfies
one of the following three conditions:
where c is an appropriate constant.
Let G2 be the graph defined over the point of P2 , connecting two points of P2 , if the diagonal rectangle they define
has area smaller than ε. Consider a point p ∈ P2 , clearly, all
its neighbors in G2 , must lie inside S(p, ε), and by the above
discussion, with high probability, the number of neighbors
of p in G2 is bounded by
C
=
O(Am log n) = O
1
ε log
ε
m log n
3
b
1. p ∈
/ ∪S.
b but there is only one region of S
b that covers
2. p ∈ ∪S,
p.
b and there exists r ∈ S \ S,
b such that p ∈ r.
3. p ∈ ∪S,
= O log n .
See Figure 3.
In particular, G2 can be colored using C colors. Let f (·)
be the coloring of P1 ∪ P2 resulting from coloring the points
Assume that we are given an algorithm A that computes, for
any set of regions S, a non-empty admissible set A(S). We
118
1111111111111
0000000000000
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
000000000
111111111
0000000000000
1111111111111
000000000
111111111
0000000000000
000000000 1111111111111
111111111
0000000000000
1111111111111
000000000
111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
000000000
111111111
00000000000
11111111111
0000000000000
1111111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
Proof. Let A = A(R) be the arrangement of the regions of R. Place an arbitrary point inside each face of the
arrangement A and let P denote the resulting point set.
Let χ be a random coloring of the regions of R, by two
colors black and white, where each region is colored independently by choosing black or white with equal probabilities.
A point p ∈ P is said to be unsafe if all the regions of R
that contain p are colored black. Let PU be the set of unsafe
points of P . Let RB be the set of all regions of R which
are colored black by χ. We construct a graph G over RB ,
connecting two regions r, r ∈ RB by an edge if there is an
unsafe point p ∈ PU that is contained inside both r and r .
Let e(G) and v(G) denote respectively, the number of
edges and of vertices in G. We claim that, with constant
probability, v(G) ≥ n/3 and e(G) = O(U (n)).
Clearly, the condition |RB | = v(G) ≥ n/3 holds with high
probability by the Chernoff inequality (see [2]). As for the
second claim, for a vertex p ∈ P , let d(p) denote the number
of regions of R that contain it. Clearly, the probability
that p is unsafe is 1/2d(p) . If p is unsafe, there are d(p)
2
pairs of regions
of RB whose intersections contain p, so p
d(p)
induces 2 edges in G. Let Xp be the random variable
d(p)
if p is unsafe. Clearly,
having value
2
P 0 if p is safe, and
e(G) ≤ p∈P Xp . Thus, using linearity of expectation and
Lemma 4.3, we have
Figure 3: A set S of discs and an admissible subset
Sb (depicted with gray).
3
can now use the algorithm A to CF-color the given regions:
(i) Compute an admissible set Sb = A(S), and assign to
all the regions in Sb the color 1. (ii) Color the remaining
regions in S \ Sb recursively, where in the ith stage we assign
the color i to the regions in the admissible set. We denote
the resulting coloring by CA (S).
E[e(G)] ≤
X
E[Xp ] =
p∈P
p∈P
d(p)>1
0
B
= O
The proof is similar to that of Lemma 2.2. We omit the
details here.
= O
Lemma 4.3 Let R be a set of n regions and let U(m) denote
the maximum complexity of the union of any m regions of R.
Let A(R) denote the arrangement of the boundary curves of
the regions in R. Then the number of faces of the arrangement A(R) that are contained inside at most k regions
of R
(denoted by F≤k (R)) is bounded by O k2 U(n/k) + n .
d(p)
2
2d(p)
1
n
X
= OB
@
Lemma 4.2 Given a set of regions S, the coloring CA (S)
is a valid conflict-free coloring of S.
X
i=2
n
X
X i C
C
2i A
2
p∈P
d(p)=i
i2
i U(n/i) · i
2
i=2
n
X
i4
i=2
!
2
U(n)
2i
!
= O (U(n)) .
Thus, by the Markov inequality, it follows that there is a
constant c, such that Pr[e(G) ≥ c · U(n)] ≤ 1/4.
It follows, that with constant probability, G has at least
n/3 vertices, and its average degree is at most 6c · U(n)/n.
Thus, by Lemma 2.3(i), G contains an independent set of
size Ω(n2 /U(n)). Let R be this independent set. It is easy
to verify that R is admissible with respect to R. Indeed let
f be a face of A(R) that is contained in at least two regions
r1 , r2 ∈ R , and let p be its representing point. Then p must
be safe, so p, and thus f , is contained also in a white region,
which clearly does not belong to R . This completes the
proof of the lemma.
Proof. Let S≤k (R) be the set of vertices of the arrangement A(R) that lie in the interior of at most k regions of
R. By the
analysis of Clarkson and Shor [5],
probabilistic
we have S≤k(R) = O k2 U(n/k) . We charge a face contained in at most k regions to its lowest vertex. Thus, the
only faces unaccountable for by this charging scheme, are
the faces that have no vertices on their boundary. However,
it is easy to check that the number of such faces is only
O(n), as we can charge such a face to the region of R that
forms its outer boundary. Thus F≤k (R) = O(S≤k (R)+n) =
O(k2 U(n/k)) + n).
Definition 4.5 ([13]) A family R of (simply connected)
Jordan regions in the plane is called a family of pseudodiscs if the boundaries if each pair of them intersect at most
twice.
Lemma 4.4 Let R be a set of n regions and let U(m) denote
the maximum complexity of the union of any m regions of
R. Then there exists an admissible
set Sb ⊆ R with respect
b = Ω n2 . Furthermore, such a set
to R, such that |S|
U (n)
Theorem 4.6 Let R be a family of n pseudo-discs. Then
R admits a CF-coloring with O(log n) colors.
R
can be computed in (expected) polynomial time.
119
Proof. The complexity of the union of any m regions of
is O(m) (see [13]).
Plugging this fact into
Lemma 4.4, we have that R contains an admissible set Sb
with respect to R of size Ω(n). Applying Lemma 4.2, and
arguing as in the proof of Lemma 2.3,we have that R admits
a CF-coloring with O(log n) colors.
A weaker deterministic version of the results in this section can be derived using discrepancy [4, 14]. However, this
yields colorings which are worse by a logarithmic factor. We
leave the question of developing a deterministic algorithm
for those problems, as an open problem for further research.
Definition 4.7 ([7]) A planar object c is (α, β)-covered if
the following holds: (i) c is simply connected, and (ii) for
any point p ∈ ∂c we can place a triangle ∆ fully inside c,
such that p is a vertex of ∆, each angle of ∆ is at least
α, and the length of each edge of ∆ is at least β times the
diameter of c.
4.1.2 Conflict Free Coloring of “Well Behaved” Unbounded Regions
In this section, we study the problem of coloring regions
with “well behaved” boundaries.
A curve γ is called an x-monotone curve, if it is the graph
of a continuous totally defined function. For an x-monotone
curve γ in the plane, let γ + (resp., γ − ) denote the set of all
points in the plane that lie above (resp., below) γ. Thus for
a straight line l, l+ (resp., l− ) is the positive (resp., negative)
half-plane which has l as its boundary.
Let R = {r1 , . . . , rn } be a family of n regions in the plane
such that each region ri is the bounded by some x-monotone
curve γi (i.e., either ri = γi+ or ri = γi− ).
Assume further that the corresponding set of curves Γ =
{γ1 , . . . , γn } has the property that each pair of curves intersect at most s times, where s is some fixed given constant.
We refer to such a family of regions as s-order half-planes.
Thus, any family of half-planes is a family of 1-order halfplanes.
Put βs (n) = λs (n)/n, where λs (n) denotes the maximum
length of a Davenport-Schinzel sequence of order s with n
symbols. It is well known that β1 (n) = 1, β2 (n) = 2 − 1/n,
β3 (n) = O(α(n)) where α(n) denotes the inverse Ackerman
function. In general βs (n) ≈ O(α(n)s−2/2 ). See [16] for
further details concerning Davenport-Schinzel sequences.
The following result is an immediate consequence of
Lemma 4.4.
Theorem 4.8 Let C be a collection of n (α, β)-covered regions in the plane, of finite description complexity, such
that the boundaries of each pair of regions of C intersect
in at most s points. Then, C has a conflict-free coloring using O(βs+2 (n) log 3 n log log n) colors, where βs+2 (n) =
λs+2 (n)/n. This coloring can be computed in expected polynomial time.
Proof. In this case, U(n) = O(λs+2 (n) log2 (n) log log n)
by the result of [7]. Thus, by Lemma 4.4, C has an admissible set of size
Ω
n2
U(n)
=
=
n2
λs+2 (n) log 2 n log log n
n
Ω
.
2
βs+2 (n) log n log log n
Ω
Applying the algorithm described in Lemma 4.2, and arguing as in Lemma 2.3, it follows that we have a conflict-free
coloring of C using
O(βs+2 (n) log3 n log log n)
colors.
Theorem 4.11 Let R = {r1 , . . . , rn } be a family of s-order
half-planes regions. There exists a CF-coloring of R with
O(βs (n) log n) colors.
4.1 CF-coloring of Simple Geometric Regions
in the Plane
In this section we study some special cases of CF-coloring
of regions in the plane, where the underlying regions need
not necessarily have low union complexity.
5. RELAXING THE NOTION OF CONFLICTFREE COLORING
4.1.1 Conflict Free Coloring of Axis-Parallel Rectangles
In this section, we generalize the notion of CF-coloring of
a range space and the notion of CF-coloring of regions.
Lemma 4.9 Let R be a set of n axis-parallel rectangles, all
intersecting the y-axis. Then there is a CF-coloring of R
with O(log n) colors.
5.1
k-CF-coloring
of a range-space
Definition 5.1 k-CF-coloring of a range-space: Let
(P, R) be a range space in Rd . A function χ : P → {1, . . . , i}
is a k-CF-coloring of (P, R) if for every r ∈ R there exists
a color j such that 1 ≤ |{p ∈ P ∩ r|χ(p) = j}| ≤ k namely,
for every possible nonempty range r there exists at least one
color j such that j appears at most k times among the colors
assigned to points of P ∩ r.
Proof. It is easy to verify that the complexity of the
union of m such rectangles is O(m). Hence, the result follows
immediately from Lemma 4.4 and Lemma 4.2.
Theorem 4.10 Let R be a set of n axis-parallel rectangles.
Then there is a CF-coloring of R using O(log2 n) colors.
Proof. Let 4 be a vertical line, such that at most n/2
rectangles of R lie fully to the left of 4, and at most n/2
rectangles of R lie fully to its right. Let R0 , R1 , R2 denote
respectively the sets of rectangles crossed by 4, lying fully
to its left, and lying fully to its right. By Lemma 4.9, we
can CF-color the set R0 with O(log n) colors. We color
recursively R1 and R2 , using the same set of colors in both
subproblems, but keeping them disjoint from the set used to
color R0 . This gives rise to a coloring of R with a total of
O(log 2 n) colors, which is easily seen to be a CF-coloring.
Let χk (P, R) denote the minimum number of colors needed
for a k-CF-coloring of (P, R).
Note that the notion of k-CF-coloring of a range space is a
generalization of the notion of CF-coloring, as a CF-coloring
is just a 1-CF-coloring.
As mentioned in Lemma 6.1, there exists range spaces
(P, R) consisting of n points in R3 , where the ranges are induced by balls for which any CF-coloring (i.e., 1-CF-coloring)
of (P, R) needs n colors. However, the problem becomes
120
k-CF coloring the given regions as follows: (i) Compute a
k-admissible set Sb = A(S) with respect to S, to and assign
all the regions in Sb the color 1. (ii) Color the remaining
regions in S \ Sb recursively, using colors ≥ 2. We denote the
resulting coloring by CA (S).
more interesting if we relax the coloring requirement to be
a k-CF-coloring with k ≥ 2.
Theorem 5.2 Let R be the set of all balls in R3 and let P
be a set of n points in R3 . Then χk (P, R) = O(n1/k ).
Proof. The proof technique is a generalization of the
ideas developed in Section 4. We define a k + 1 uniform
hypergraph H = (P, E ) as follows. E is the collection of all
subsets of P of size k + 1 that are realizable by a range in
R. By the Clarkson-Shor technique it is easy to see that
|E| = O(k2 n2 ). Thus the average degree of H is O(k2 n)
and therefor by Lemma 2.5 there exists an independent set
P ⊂ P of size Ω(n1−1/k ) (note that independency means
that any ball that contains at least k + 1 points of P , must
also contain a point from P \ P ). We can color all points
of P by a single color, say 1, and iterate on P \ P . Thus,
the total number of colors we use is O(n1/k ). It is easy
to see (similar to Lemma 2.2) that this coloring is a valid
k-CF-coloring of (P, R).
5.2
k-CF-coloring
Lemma 5.5 Given a set of regions S, the coloring CA (S)
is a valid k-conflict-free coloring of S.
The proof is similar to that of Lemma 2.2 and Lemma 4.2.
The following result extends Lemma 4.3 to three dimensions.
Lemma 5.6 Let R be a set of n regions in R3 and let U(m)
denote the maximum complexity of the boundary surfaces
of the union of any m regions of R. Then the number of
3-dimensional cells of the arrangement A(R) that are contained in at most i regions of
R (denoted by F≤i (R) ) is
bounded by O i3 U(n/i) + n2 .
Proof. Let S≤i (R) be the set of vertices of the arrangement A(R) that lie in the interior of at most i regions of
R. By the Clarkson-Shor
technique [5], we have |S≤i (R)| =
O i3 U(n/i) . We charge a cell contained in at most i regions, to its lowest vertex. Thus, the only cells unaccountable for by this charging scheme are the cells that have no
vertices on their boundary. However, it is easy to check
that the number of such cells is bounded by O(n2 ). Thus
F≤i (R) = O(S≤i (R) + n2 ).
of regions
Definition 5.3 k-CF-coloring of regions: Let R be a
collection of regions in Rd . A function χ : R → {1, . . . , i}
is a k-CF-coloring of R if for every point p ∈ ∪r∈R r there
exists a color j such that 1 ≤ |{r ∈ R|p ∈ r, χ(r) = j}| ≤ k
that is, for every possible point p in the union of R there
exists at least one color j such that j appears at most k
times among the colors assigned to the regions of R that
contain p.
Lemma 5.7 Let R be a set of n regions in R3 , and let U(m)
denote the maximum complexity of the union of any m reb
gions of R. Then there exists a k-admissible
1+1/k set S ⊆ R with
n
b = Ω
. Furthermore,
respect to R, such that |S|
U (n)1/k
such a set can be computed in (expected) polynomial time.
Note that the notion of k-CF-coloring of regions is a generalization of the notion of CF-coloring, as a CF-coloring is
just a 1-CF-coloring.
We focus on k-CF-coloring of balls in R3 . Note that the
union of a set of n balls can have Ω(n2 ) complexity and
one cannot apply the technique developed in Section 4 to
obtain non-trivial bounds on the number of colors needed
for 1-CF-coloring of such a set of regions. However, as we
will show in this section, one can obtain non-trivial bounds
on the number of colors needed for k-CF-coloring of set of
regions in R3 with near quadratic union complexity, for k a
pre-specified parameter (k ≥ 2). The approach that we use
generalizes to any fixed dimension.
Let R be a family of regions in R3 , such that the complexity of union of any n regions of R is at most O(U(n)).
In the following, we assume that U(n) is a near quadratic
function. This holds for balls.
Proof. The proof follows closely the ideas of the proof
of Lemma 4.4 with a slight twist. Let A = A(R) be the
arrangement of the (boundary surfaces of the) regions of R.
Place an arbitrary point inside each (three-dimensional) cell
of the arrangement A and let P denote the resulting point
set.
Let χ be a random coloring of the regions of R, by two
colors black and white, where each region is colored independently by choosing black or white with equal probabilities.
A point p ∈ P is said to be unsafe if all the regions of R
that contain p are colored black. Let PU be the set of unsafe points of P . Let RB be the set of all regions of R
which are colored black by χ. We construct a k + 1-uniform
hyper-graph H over RB , such that k + 1 tuple of regions
r1 , . . . , rk+1 ∈ RB form a hyperedge if there is an unsafe
point p ∈ PU that is contained inside ∩k+1
j=1 rj .
Let e(H) and v(H) denote respectively, the number of hyperedges and of vertices of H. We claim that, with constant
probability, v(H) ≥ n/3 and e(H) = O(U (n)).
Clearly, the condition |RB | = v(H) ≥ n/3 holds with
high probability by the Chernoff inequality (see, e.g., [2]).
Similarly to the proof of Lemma 4.4 the probability
that p is
unsafe is 1/2d(p) . If p is unsafe, there are d(p)
(k+1)-tuples
k+1
of regions
of RB whose intersection contain p so p induces
d(p)
hyperedges in H. Let Xp be the random variable
k+1
having value 0 if p is safe, and d(p)
if p is unsafe. Clearly,
k+1
Definition 5.4 For a set S of n regions of R, a subset Sb ⊆
S is k-admissible with respect to S, if any p ∈ R3 satisfies
one of the following three conditions:
b
1. p ∈
/ ∪S.
b but there are at most k regions of S
b that cover
2. p ∈ ∪S,
p.
b and there exists r ∈ S \ S,
b such that p ∈ r.
3. p ∈ ∪S,
Assume that we are given an algorithm A that computes
for any set S of regions a non-empty k-admissible set A(S)
with respect to S. We can then use the algorithm A for
121
P
Proof. Take P to be a set of n points on the positive
branch of the moment curve γ = (t, t2 , t3 )|t ≥ 0 in R3 .
It is easy to verify that any pair of points p, q ∈ P are
connected in the Delaunay triangulation of P [9], implying
that there exists a ball hose intersection with P is {p, q}.
Thus, all points must be colored using different colors.
The second claim follows by lifting P into the standard
paraboloid in R4 by the map (x, y, z) %→ (x, y, z, x2 +y 2 +z 2 ).
A ball in R3 is mapped to halfspace in R4 so that a point p
lies in the ball if and only if its image lies in the halfspace.
It follows, that n colors are necessary in any CF-coloring of
the image of P .
e(H) ≤ p∈P Xp . Thus, using linearity of expectation and
Lemma 5.6, we have
X
E[e(h)] ≤
p∈P
B
OB
@
=
1
n
X
i=k+1
=
X ik+1 C
C
2i A
i=k+1 p∈P
d(p)=i
n
X
O
ik+1
i U(n/i) · i
2
i=k+1
!
3
n
X
ik+4
O
d(p)
k+1
2d(p)
p∈P
d(p)≥k+1
0
=
X
E[Xp ] =
2i
!
U(n)
Observation 6.2 The conflict-free chromatic number of the
range space of points and fat triangles in the plane is n. Indeed, place n points along a circle. It is easy to verify that
for every pair of points, one can construct a fat triangle that
contains the two points, but does not contain any other point.
Hence, as above, n colors are needed.
= O (U(n)) .
Thus, by the Markov inequality, it follows that there is a
constant c, such that Pr[e(H) ≥ c · U(n)] ≤ 1/4.
It follows, that with constant probability, H has at least
n/3 vertices, and its average degree is at most (k + 1)3c ·
U(n)/n. Thus, by Lemma 2.5(ii), H contains
an indepen
dent set of size Ω(
n
(
U(n) 1/k
)
n
)=Ω
n1+1/k
U (n)1/k
Note, that there is no direct connection between the VCdimension of the ranges used, and their conflict-free chromatic number, as demonstrated by Lemma 6.1 (and Observation 6.2), showing a range-space that requires n colors and
has a finite VC-dimension.
. Let R be this
independent set. It is easy to verify that R is k-admissible
with respect to R. This completes the proof of the lemma.
Note that when U(n) = O(n2 ) we have a k-admissible set of
size Ω(n1−1/k )
6.1 Adding Points
Lemma 6.3 Given a set P of n points in the plane, and
R the set of all rectangles, one can compute a set P of
O(n log n) points, such that kopt (P ∪ P , R) = O(log2 n).
Theorem 5.8 Let R be a set of n balls in R3 . There exists
a k-CF-coloring of R with a total of at most O(n1/k ) colors.
Proof. By Lemma 5.7 there exists a k-admissible set R
with respect to R of size Ω(n1−1/k ). Plugging this fact to the
algorithm suggested by Lemma 5.5 completes the proof.
Proof. Let 4 be a ’median’ vertical line of P , and let
S(P ) be the set of points resulting from the horizontal projection of the points of P into 4. Let Pl (4) and Pr (4) be
the subset of points of P to the left and to the right of 4,
respectively. Let P = S(P ) ∪ L ∪ R, where L and R and
the set of points resulting from applying the above process
to Pl (4) and Pr (4), respectively.
Clearly, |P | = O(n log n), and observe that any rectangle
of P that contains two points of P , must contain a point
of P . Furthermore, the points of P can be colored using O(log 2 n) colors, by coloring the points of S(P ) using
O(log n) colors, by Lemma 3.5, and recursing on L and R.
(Note, that we can color all the points of P by the same
color.)
Remark 5.9 A closer look into the analysis of the proof of
Lemma 5.7 reveals the fact that the lemma generalizes to
any dimension.
We thus have:
Theorem 5.10 Let R be a set of n regions in Rd with the
property that the complexity of the union of any m regions
of R is at most U(m). Then, there exists
a k-admissible
1+1/k set Sb ⊆ R with respect to R, such that Sb = Ω Un(n)1/k .
Furthermore, such a set can be computed in (expected) polynomial time.
7. CONCLUSIONS
We proved several results on CF-coloring of points and
regions. There are numerous problems for further research
suggested by our results. In particular, the main open problems we pose in this paper are:
Theorem 5.11 Let R be a set of n regions in Rd with the
property that the complexity of the union of any m regions of
R is at most U(m) = O(m1+α ), where 0 < α. Then, there
exists a k-CF-coloring of R with O(nα/k ) colors.
1. Improve the bounds on the k-CF-chromatic number
of points, with respect to balls, in Rd , for d > 2 and
k ≥ 2.
.
6.
MISCELLANEOUS
2. Improve the bounds on the number of colors needed for
k-CF-coloring of n balls in Rd , for d > 2 and k ≥ 2.
Lemma 6.1 Let R be the set of balls in three dimensions.
Then kopt (P, n) = n.
The same holds where R is the set of halfspaces in Rd , for
d > 3.
3. Improve the bounds on the CF-chromatic number of
points in the plane with respect to axis-parallel rectangles.
122
Acknowledgments
[9] J. Erickson. Nice point sets can have nasty delaunay
triangulations. In Proc. 17th Annu. ACM Sympos.
Comput. Geom., pages 96–105, 2001.
[10] G. Even, Z. Lotker, D. Ron, and S. Smordinsky.
Conflict-free colorings of simple geometric regions
with applications to frequency assignment in cellular
networks. In Proc. 43th Annu. IEEE Sympos. Found.
Comput. Sci., 2002. to appear.
[11] S. Har-Peled and S. Smorodinsky. On conflict-free
coloring of points and simple regions in the plane.
http://www.uiuc.edu/˜sariel/research/papers/02/
coloring, 2003.
[12] D. Haussler and E. Welzl. ε-nets and simplex range
queries. Discrete Comput. Geom., 2:127–151, 1987.
[13] K. Kedem, R. Livne, J. Pach, and M. Sharir. On the
union of Jordan regions and collision-free translational
motion amidst polygonal obstacles. Discrete Comput.
Geom., 1:59–71, 1986.
[14] J. Matoušek. Geometric Discrepancy. Springer, 1999.
[15] J. Pach and G. Tóth. Conflict free colorings. Discrete
Comput. Geom., 2003. to appear (The
Goodman-Pollack festschrift.).
[16] M. Sharir and P. K. Agarwal. Davenport-Schinzel
Sequences and Their Geometric Applications.
Cambridge University Press, New York, 1995.
[17] S. Smorodinsky. Combinatorial problems in
computational geometry. Ph.D dissertation, Tel-Aviv
University, in preparation, 2003.
[18] D. B. West. Intorudction to Graph Theory. Prentice
Hall, 2ed edition, 2001.
The authors would like to thank Pankaj Agarwal, Noga
Alon, Boris Aronov, Timothy Chan, Jeff Erickson, Guy Even,
János Pach, Rom Pinchasi and Micha Sharir for helpful discussions of the problems studied in this paper. In particular,
the authors would like to thank Micha Sharir for his help
with the manuscript.
8.
REFERENCES
[1] N. Alon, M. Krivelevich, and B. Sudakov. Coloring
graphs with sparse neighborhoods. J. Comb. Theo.
Ser. B, 77(73–82), 1998.
[2] N. Alon and J. H. Spencer. The probabilistic method.
Wiley Inter-Science, 2nd edition, 2000.
[3] C. Berge. Hypergraphs. North Holland, Amsterdam,
1989.
[4] B. Chazelle. The Discrepancy Method. Cambridge
University Press, 2000.
[5] K. L. Clarkson and P. W. Shor. Applications of
random sampling in computational geometry, II.
Discrete Comput. Geom., 4:387–421, 1989.
[6] M. de Berg, M. van Kreveld, M. H. Overmars, and
O. Schwarzkopf. Computational Geometry: Algorithms
and Applications. Springer-Verlag, 2nd edition, 2000.
[7] A. Efrat. The complexity of the union of
(α, β)-covered objects. In Proc. 15th Annu. ACM
Sympos. Comput. Geom., pages 134–142, 1999.
[8] A. Efrat and M. Sharir. On the complexity of the
union of fat convex objects in the plane. Discrete
Comput. Geom., 23:171–189, 2000.
123