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Online conflict-free colorings for hypergraphs
Amotz Bar-Noy∗
Panagiotis Cheilaris†
Svetalana Olonetsky‡
Shakhar Smorodinsky§
Abstract
We provide a framework for online conflict-free coloring (CF-coloring) of any hyper-
graph.We use this framework to obtain an efficient randomized online algorithm for
CF-coloring any k-degenerate hypergraph. Our algorithm uses O(klogn) colors with high
probability and this bound is asymptotically optimal for any constant k. Moreover, our al-
gorithm uses O(klogklogn) random bits with high probability. As a corollary, we obtain
asymptotically optimal randomized algorithms for online CF-coloring some hypergraphs
that arise in geometry. Our algorithm uses exponentially fewer random bits compared to
previous results.
We introduce deterministic online CF-coloring algorithms for points on the line with
respect to intervals and for points on the plane with respect to halfplanes (or unit discs)
that use Θ(logn) colors and recolor O(n) points in total.
1Introduction
A hypergraph is a pair (V,E), where V is a finite set and E ⊂ 2V. The set V is called the
ground set or the vertex set and the elements of E are called hyperedges. A proper k-coloring of
a hypergraph H = (V,E), for some positive integer k, is a function χ : V → {1,2,...,k} such
that no S ∈ E with |S| ≥ 2 is monochromatic. Let χ(H) denote the minimum integer k for
which H has a k-coloring. χ(H) is called the chromatic number of H. A conflict-free coloring
(CF-coloring) of H is a coloring of V with the further restriction that for any hyperedge
S ∈ E there exists a vertex v ∈ S with a unique color (i.e., no other vertex of S has the
same color as v). Both proper coloring and CF-coloring of hypergraphs are generalizations of
vertex coloring of graphs (the definition coincides when the underlying hypergraph is a simple
graph). Therefore the computational complexity of such colorings is at least as hard as for
simple graphs.
The study of conflict-free colorings was originated in the work of Even et al. [5] and
Smorodinsky [13] who were 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. Base stations are interconnected by
an external fixed backbone network whereas clients are connected only to base stations. Con-
nections between clients and base stations are implemented by radio links. Fixed frequencies
∗Computer and Information Science Department, Brooklyn College, 2900 Bedford Avenue Brooklyn,
NY, 11210. E-mail: amotz@sci.brooklyn.cuny.edu.
†The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY, 10016.
E-mail: philaris@sci.brooklyn.cuny.edu.
‡Tel-Aviv University, E-mail: olonetsk@post.tau.ac.il.
§Courant Institute for Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012.
E-mail: shakhar@cims.nyu.edu.
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are assigned to base stations to enable links to clients. Clients continuously scan frequencies
in search of a base station with good reception. The fundamental problem of frequency as-
signment in such cellular networks is to assign frequencies to base stations so that every client,
located within the receiving range of at least one station, can be served by some base station,
in the sense that the client is located within the range of the station and no other station
within its reception range has the same frequency (such a station would be in “conflict” with
the given station due to mutual interference). The goal is to minimize the number of assigned
frequencies (“colors”) since the frequency spectrum is limited and costly.
Suppose we are given a set of n base stations, also referred to as antennas. Assume, for
simplicity, that the area covered by a single antenna is given as a disk in the plane. Namely, the
location of each antenna and its radius of transmission is fixed and is given (the transmission
radii of the antennas are not necessarily equal). Even et al. [5] showed that one can find an
assignment of frequencies to the antennas with a total of at most O(logn) frequencies such that
each antenna is assigned one of the frequencies and the resulting assignment is free of conflicts,
in the preceding sense. Furthermore, it was shown that this bound is worst-case optimal. Let
R be a set of regions in the plane. For a point p ∈ ∪r∈Rr, let r(p) = {r ∈ R | p ∈ r}. Let H(R)
denote the hypergraph (R,{r(p) | p ∈ ∪r∈R}). We say that H(R) is the hypergraph induced
by R. Thus, Even et al. [5] showed that any hypergraph induced by a family R of n discs
in the plane admits a CF-coloring with only O(logn) colors and that this bound is tight in
the worst case. Furthermore, such a coloring can be found in deterministic polynomial time1.
The results of [5] were further extended in [8] by combining more involved probabilistic and
geometric ideas. The main result of [8] is a general randomized 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 addition to the practical motivation, this new coloring model has
drawn much attention of researchers through its own theoretical interest and such colorings
have been the focus of several recent papers (see, e.g., [4, 5, 6, 8, 10, 12, 13, 14]). To capture
a dynamic scenario where antennas can be added to the network, Fiat et al. [6] initiated the
study of online CF-coloring of hypergraphs. They considered a very simple hypergraph H
which has its vertex set represented as a set P of n points on the line and its hyperedge set
consists of all intersections of the points with some interval. The set P ⊂ R is revealed by an
adversary online: Initially, P is empty, and the adversary inserts points into P, one point at a
time. Let P(t) denote the set P after the t-th point has been inserted. Each time a point is
inserted, the algorithm needs to assign a color c(p) to it, which is a positive integer. Once the
color has been assigned to p, it cannot be changed in the future. The coloring should remain
conflict-free at all times. That is, for any interval I that contains points of P(t), there is a color
that appears exactly once in I. Among other results, [6] provided a randomized algorithm for
online CF-coloring n points on the line with O(lognloglogn) colors with high probability.2
They also provided a deterministic algorithm for online CF-coloring n points on the line with
Θ(log2n) colors in the worst case.
An online CF-coloring framework:
of online CF-coloring applied to arbitrary hypergraphs. Suppose the vertices of an underlying
hypergraph H = (V,E) are given online by an adversary. Namely, at every given time step t, a
new vertex vt∈ V is given and the algorithm must assign vta color such that the coloring is a
1In [5] it is shown that finding the minimum number of colors needed to CF-color a given collection of discs
is NP-hard even when all discs are congruent, and an O(logn) approximation-ratio algorithm is provided.
2Their algorithm assumes that the adversary is oblivious in the sense that it does not have access to the
random coin flips of the algorithm.
In this paper, we investigate the most general form
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valid conflict-free coloring of the hypergraph that is induced by the vertices Vt= {v1,...,vt}
(see the exact definition in section 2). Once vtis assigned a color, that color cannot be changed
in the future. The goal is to find an algorithm that minimizes the maximum total number of
colors used (where the maximum is taken over all permutations of the set V ).
We present a general framework for online CF-coloring any hypergraph. Interestingly, this
framework is a generalization of some known coloring algorithms. For example the Unique-
Max Algorithm of [6] can be described as a special case of our framework. Also, when the
underlying hypergraph is a simple graph then the First-Fit online algorithm is another special
case of our framework.
Based on this framework, we introduce a randomized algorithm and show that the maxi-
mum number of colors used is a function of the ‘degeneracy’ of the hypergraph. We define the
notion of a k-degenerate hypergraph as a generalization of the same notion for simple graphs.
Specifically we show that if the hypergraph is k-degenerate, then our algorithm uses O(klogn)
colors with high probability. This is asymptotically tight for any constant k.
As demonstrated in [6], the problem of online CF-coloring the very special hypergraph
induced by points on the real line with respect to intervals is highly non-trivial. The best
randomized online CF-coloring algorithm of [6] uses O(lognloglogn) colors. Chen, Kaplan
and Sharir [10] studied the special hypergraph induced by points in the plane with respect
to halfplanes and unit discs and obtained a randomized online CF-coloring with O(log3n)
colors with high probability. Recently, the bound Θ(logn) just for these two special cases was
obtained independently by Chen [3]. Our algorithm is more general and uses only Θ(logn)
colors; an interesting evidence to our algorithm being fundamentally different from the ones in
[3, 6, 10], when used for the special case of hypergraphs that arise in geometry, is that it uses
exponentially fewer random bits. The algorithms of [3, 10] use Θ(nlogn) random coin flips
and our algorithm uses O(logn) random coin flips.
Another interesting corollary of our result is that we obtain a randomized online coloring for
k-inductive graphs with O(klogn) colors with high probability. This case was studied by Irani
[9] who showed that a greedy First-Fit algorithm achieves the same bound deterministically.
Deterministic online CF-coloring with recoloring:
coloring where at each step, in addition to the assignment of a color to the newly inserted point,
we allow some recoloring of other points. The bi-criteria goal is to minimize the total number
of recolorings done by the algorithm and the total number of colors used by the algorithm.
We introduce an online algorithm for CF-coloring points on the line with respect to intervals,
where we recolor at most one already assigned point at each step. Our algorithm uses Θ(logn)
colors. This is in contrast with the O(log2n) colors used by the best known deterministic
algorithm by Fiat et al. [6] that does not recolor points. We also show online algorithm for
CF-coloring points on the plane with respect to halfplanes that uses Θ(logn) colors and the
total number of recolorings is O(n). For this problem no deterministic algorithm was known
before.
We initiate the study of online CF-
Paper organization:
presents the general framework for online CF-coloring of hypergraphs. Section 4 introduces
the randomized algorithm derived from the framework. Section 5 shows deterministic online
algorithm for intervals and halfplanes with recoloring. Section 6 describes the results for the
hypergraphs that arise from geometry. Finally, Section 7 concludes with a discussion and some
open problems.
Section 2 defines the notion of a k-degenerate hypergraph. Section 3
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2 Preliminaries
We start with some basic definitions:
Definition 2.1. Let H = (V,E) be a hypergraph. For a subset V?⊂ V let H(V?) be the
hypergraph (V?,E?) where E?= {e ∩ V?|e ∈ E}. H(V?) is called the induced hypergraph on V?.
Definition 2.2. For a hypergraph H = (V,E), the Delaunay graph G(H) is the simple graph
G = (V,E) where the edge set E is defined as E = {(x,y) | {x,y} ∈ E} (i.e., G is the graph
on the vertex set V whose edges consist of all hyperedges in H of cardinality two).
Definition 2.3. A simple graph G = (V,E) is called k-degenerate (or k-inductive) for some
positive integer k, if every (vertex-induced) subgraph of G has a vertex of degree at most k.
Definition 2.4. Let k > 0 be a fixed integer and let H = (V,E) be a hypergraph on n vertices.
Fix a subset V?⊂ V . For a permutation π of V?such that V?= {v1,...,vi} (where i = |V?|)
let Cπ(V?) =?i
Assume that ∀V?⊂ V and for all permutations π ∈ S|V?|we have Cπ(V?) ≤ k|V?|. Then we
say that H is k-degenerate.
j=1d(vj), where d(vj) = |{l < j|(vj,vl) ∈ G(H({v1,...,vj}))}|, that is, d(vj) is
the number of neighbors of vjin the Delaunay graph of the hypergraph induced by {v1,...,vj}.
It is easy to see that our definition of a k-degenerate hypergraph is a generalization of that
of a k-degenerate graph.
3An online CF-coloring framework
Let H = (V,E) be any hypergraph. Our goal is to define a framework that colors the vertices
V in an online fashion. That is, the vertices of V are revealed by an adversary one at a time.
At each time step t, the algorithm must assign a color to the newly revealed vertex vt. This
color cannot be changed in the future. The coloring has to be conflict-free for all the induced
hypergraphs H(Vt) t = 1,...,n, where Vt⊂ V is the set of vertices revealed by time t.
For a fixed positive integer h, let A = {a1,...,ah} be a set of h auxiliary colors (not to be
confused with the set of ‘real’ colors used for the CF-coloring: {1, 2, ...}). Let f : N → A be
some fixed function. We now define the framework that depends on the choice of the function
f and the parameter h.
A table (to be updated online) is maintained where each entry i at time t is associated
with a subset Vi
colors. We say that f(i) is the color that represents entry i in the table. At the beginning all
entries of the table are empty. Suppose all entries of the table are updated until time t − 1
and let vt be the vertex revealed by the adversary at time t. The framework first checks if
an auxiliary color can be assigned to vtsuch that the auxiliary coloring of V1
the color of vtis a proper coloring of H(V1
used by the framework. For example a first-fit greedy in which all colors in the order a1, ...,
ahare checked until one is found. If such a color cannot be found for vt, then entry 1 is left
with no changes and the process continues to the next entry. If however, such a color can be
assigned, then vtis added to the set V1
vt. If this color is the same as f(1) (the auxiliary color that is associated with entry 1), then
the final color in the online CF-coloring of vtis 1 and the updating process for the t-th vertex
stops. Otherwise, if an auxiliary color cannot be found or if the assigned auxiliary color is not
the same as the color associated with this entry, the updating process continues to the next
t⊂ Vtin addition to an auxiliary proper coloring of H(Vi
t) with at most h
t−1together with
t−1∪ {vt}). Any (proper) coloring procedure can be
t−1. Let c denote such an auxiliary color assigned to
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entry. The updating process stops at the first entry i for which vtis both added to Vi
the auxiliary color assigned to vtis the same as f(i). The color of vtin the final conflict-free
coloring is then set to i.
It is possible that vtnever gets a final color. In this case we say that the framework does not
halt. However, termination can be guaranteed by imposing some restrictions on the auxiliary
coloring method and the choice of the function f. For example, if first-fit is used for the
auxiliary colorings at any entry and if f is the constant function f(i) = a1, for all i, then the
framework is guaranteed to halt for any time t. An example instantiation of the framework
for conflict-free coloring with respect to intervals is given in the appendix. In section 4 we
derive a randomized online algorithm based on this framework. This algorithm always halts3
and moreover it halts after a “small” number of entries with high probability.
We now turn to prove that the above framework produces a valid CF-coloring in case it
halts.
tand
Lemma 3.1. If the above framework halts for any vertex vt then it produces a valid online
CF-coloring of H.
Proof. See appendix.
The above algorithmic framework can also describe some well-known deterministic algo-
rithms. For example, if first-fit is used for auxiliary colorings and f is the constant function,
f(i) = a1, for all i, then: (a) If the input hypergraph is induced by points on a line with
respect to intervals as in example 1 then the algorithm derived from the framework becomes
identical to the Unique Maximum Greedy algorithm described and analyzed in [6]. (b) If the
input is a k-degenerate graph (also called k-inductive graph), the derived algorithm is identi-
cal to the First-Fit greedy algorithm for coloring graphs online. The performance of First-Fit
for restricted classes of graphs has been analyzed in several papers [7, 9, 11]. Especially for
k-inductive graphs, the First-Fit algorithm is analyzed in [9], where it is proved that it uses
O(klogn) colors.
4 An online randomized CF-coloring algorithm
There is a randomized online CF-coloring in the oblivious adversary model that always pro-
duces a valid coloring and the number of colors used is related to the degeneracy of the
underlying hypergraph in a manner described in theorem 4.1.
Theorem 4.1. Let H = (V,E) be a k-degenerate hypergraph on n vertices. Then there exists
a randomized online CF-coloring for H which uses at most O(log1+
with high probability.
1
4k+1n) = O(klogn) colors
The algorithm is based on the framework of section 3. In order to define the algorithm, we
need to state what is the function f, the set of auxiliary colors of each entry and the algorithm
we use for the auxiliary coloring at each entry. We use the set A = {a1,...,a2k+1}. For each
entry i, the representing color f(i) is chosen uniformly at random from A. We use a first-fit
algorithm for the auxiliary coloring.
Our assumption on the hypergraph H (being k-degenerate) implies that at least half of the
vertices up to time t that ‘reached’ entry i (but not necessarily added to entry i), denoted by
3It is easy to prove that in the randomized version the algorithm will halt with probability 1, but we omit
the discussion of this issue here
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Xt
easily implied by the fact that at least half of those vertices vthave at most 2k neighbors in
the Delaunay graph of the hypergraph induced by Xt−1
at most k??Xt
these ‘free’ colors then the updated coloring is indeed a proper coloring of the corresponding
induced hypergraph as well.
≥i, have been actually given some auxiliary color at entry i (that is,
??Vi
t
??≥1
2
??Xt
≥i
??). This is
≥i
(since the sum of these quantities is
≥i
??and since Vi
t⊂ Xt
≥i). Therefore since we have 2k +1 colors available, there is
always a free color to assign to such a vertex. The following lemma shows that if we use one of
Lemma 4.2. Let H = (V,E) be a k-degenerate hypergraph and let Vj
time t and at level j as produced by the above algorithm. Then for any j and t if vtis assigned
a color distinct from all its neighbors in the Delaunay graph G(H(Vj
with the colors assigned to the vertices Vj
tbe the subset of V at
t)) then this color together
t−1is also a proper coloring of the hypergraph H(Vj
t).
Proof. See appendix.
We proceed to the analysis of the performance of the algorithm.
Lemma 4.3. Let H = (V,E) be a hypergraph and let χ be a coloring produced by the above
algorithm on an online input V = {vt} for t = 1,...,n. Let Xi(respectively X≥i) denote the
random variable counting the number of points of V that were assigned a final color at entry
i (respectively a final color at some entry ≥ i). Let Ei= E[Xi] and E≥i= E[X≥i] (note that
X≥i+1= X≥i− Xi). Then:
E≥i≤
4k + 2
Proof. By induction on i. The case i = 1 is trivial. Assume that the statements hold for i.
To complete the induction step, we need to prove that E≥i+1≤ (4k+1
expectation formula, we have for any two random variables X, Y that E[X] = E[E[X | Y ]].
Thus
?4k + 1
?i−1
n
4k+2)in. By the conditional
E≥i+1= E[E[X≥i+1| X≥i]] = E[E[X≥i− Xi| X≥i]] = E[X≥i− E[Xi| X≥i]].
It is easily seen that E[Xi| X≥i] ≥1
auxiliary color by the above algorithm. Moreover each of those elements that got an auxiliary
color had probability
is oblivious and does not have access to the random bits) to get the final color i. Thus
2
X≥i
2k+1since at least half of the vertices of X≥igot an
1
2k+1(This is the only place where we need to assume that the adversary
E[X≥i− E[Xi| X≥i]] ≤ E[X≥i−
1
2(2k + 1)X≥i] =4k + 1
4k + 2E[X≥i] ≤
?4k + 1
4k + 2
?i
n
by linearity of expectation and by the induction hypotheses. This completes the proof of the
lemma.
Lemma 4.4. The expected number of colors used by the above algorithm is at most log 4k+2
4k+1n+1.
Proof. Let Iibe the indicator random variable for the following event: some points are colored
with a real color in entry i. We are interested in the number of colors used, that is Y :=?∞
E[Y ] = E[
Ii] = E[
1≤i≤b(k,n)
by Markov’s inequality and lemma 4.3.
i=1Ii.
Let b(k,n) = log(4k+2)/(4k+1)n. Then,
?
1≤i
?
Ii] + E[X≥b(k,n)+1] ≤ b(k,n) + 1
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We also have the following concentration result:
Pr[more than cb(k,n) colors are used] = Pr[X≥cb(k,n)+1≥ 1] ≤ E≥cb(k,n)+1≤
1
nc−1
by Markov’s inequality and by lemma 4.3.
This completes the performance analysis of our algorithm.
Remark: In the above description of the algorithm, all the random bits are chosen in advance
(by deciding the values of the function f in advance). However, one can be more efficient and
calculate the entry f(i) only at the first time we need to update entry i, for any i. Since at each
entry we need to use O(logk) random bits and we showed that the number of entries used is
O(klogn) w.h.p then the total number of random bits used by our algorithm is O(klogklogn)
w.h.p.
5 Deterministic online CF-coloring algorithm with recoloring
5.1 An O(logn) algorithm with recoloring for intervals
We describe a deterministic online CF-coloring algorithm for intervals that is only allowed to
recolor a single “old” point during each insertion of a new point. The algorithm is based on the
framework developed in Section 3 where we use 3 auxiliary colors {a,b,c} and f is the constant
function f(l) = a ∀l. We refer to points colored with b or c as d-points. In order to have only
logarithmic number of entries, we slightly modify the framework (using a recoloring procedure)
such that the number of points colored with a in each entry of the table is at least one third
of the total points that reach that entry. To achieve this goal, our algorithm maintains the
following invariant in every level: There are at most two d-points between every pair of points
colored with a (i.e., between every pair that are consecutively colored a among the a-points).
Therefore, at least a third of the points at each entry get color a, and two thirds are deferred
for coloring in a higher entry. The total number of colors is at most log3/2n+1. When a new
point p arrives, it is colored according to the following algorithm, starting from entry 1:
• If p is not adjacent to a point colored with an auxiliary color a then p is assigned auxiliary
color a and gets its final color in that entry.
• We color point p with b or c greedily as long as it does not break the invariant that
between any two consecutive a’s we have at most two d-points.
• It remains to handle the case where the new point p has a point colored with a on one
side and a point, say q, colored with d on the other side, such that q has no adjacent point
colored with a. We assign to p the auxiliary color of q (thus it is a d-point) in the current
entry and in all higher entries for which q got an auxiliary color and assign to it the
real color of q, and we recolor q with the auxiliary color a (and delete the corresponding
appearance of it in all higher entries of he table), and thus we recolor q with the real
color of the current entry. At this point all points have an assignment of real colors. It
is not hard to check that when we recolor a point then we do not violate the invariants
at any entry: Let ? be the entry that caused recoloring, all entries before it remain the
same, the change in the entry ? does not break invariants, all other entries remain the
same except that point p appears there instead of point q that was there before and there
are no points between p and q that appear in an entry higher than ?.
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It can be easily checked that this algorithm produces a valid CF-coloring. Also, it can be
proved that the number of recolorings is at most n−(?lgn?+1), and this is tight. An example
of a run of the algorithm is shown in Figure 2, in the appendix.
5.2 An O(logn) recoloring algorithm for circular arcs
We define a hypergraph H closely related to the one induced by intervals: The vertex set of
H is represented as a set P of n distinct points on a circle C and its hyperedge set consists
of all intersections of the points with some circular arc of C. In the static case, it can be
shown that n points can be optimally conflict-free colored with respect to circular arcs with
?lg(n − 1)? + 2 colors. Here, we are interested in an online setting, where the set P ⊂ C is
revealed incrementally by an adversary, and, as usual, the algorithm has to commit on a color
for each point without knowing how future points will be requested. Algorithms for intervals
can be used almost verbatim for circular arcs. In fact, the recoloring algorithm for intervals,
given in section 5.1, can be used verbatim, if the notion of adjacency of points is adapted to
the closed curve setting (for n ≥ 3, each point has exactly 2 immediate neighboring points,
whereas in the intervals case, the two extreme points have only one neighbor). Again, in each
entry ?, at least a third of the points is assigned auxiliary color a, and thus at most log3/2n+1
colors are used.
5.3 An O(logn) recoloring algorithm for circular arcs with substitution of
points
We consider a variation on the problem of online conflict-free coloring with respect to circular
arcs that was given in section 5.2. In this new variation, the adversary has, in addition to the
‘insertion move’ of a new point, a ‘substitution move’:
The adversary can substitute a set Q of already requested consecutive points with
a single new point p, and the algorithm has to color p, such that the whole set of
points is conflict-free colored with respect to circular arcs (in that set, p is included,
but all points in Q are removed).
Our algorithm for this variation of the problem relies on the one given in section 5.2. For an
insertion move of the adversary, it colors the new point like in section 5.2. For a substitution
move of the adversary, it colors the new point p, with the highest color occurring in the points
of Q. Point p also gets the entries of the unique point q ∈ Q with the highest color. It is
not difficult to see that the coloring remains conflict-free after each move. We remark that a
recoloring can happen only in an insertion move and that substitution moves do not make the
algorithm introduce new colors. The following is true for every t:
Lemma 5.1. After t moves, the above coloring algorithm uses at most log3/2t + 1 colors.
Proof. During a substitution move we might break the invariant that between any pair of
consecutive a’s there are at most two d-points. However if we denote in each entry a point
colored with a which was deleted to get the final color ∗ by a∗, then it is easy to see that
between any two consecutive points colored with a or a∗, there are at most two d-points and
thus it implies that at least one third of the points in every level are colored either by that
level or by ∗. We omit the easy details of the proof here.
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5.4 An O(logn) algorithm with recoloring with respect to halfplanes
In this section we describe a deterministic algorithm for online CF-coloring points with respect
to halfplanes that uses O(logn) colors and recolors O(n) points. Thus it can also be modified
for CF-coloring points in the plane with respect to unit discs as remarked in Section 6. At
every time instance t, the algorithm maintains the following invariant (Vtis the set of points
that have appeared): All points (strictly) inside the convex hull of Vt are colored with the
same special color, say ‘?’. The set of points on the convex hull of Vt, denoted by CH(Vt), are
colored with another set of colors, such that every set of consecutive points on the convex hull
has a point with a unique color. If one considers the points of CH(Vt) in the circular order
they appear on the convex hull, it is enough to CF-color them (with that circular order) with
respect to circular arcs. The number of colors used in CH(Vt) must be logarithmic on t. It is
not difficult to see that every subset of points of Vtinduced by a halfplane contains a set of
consecutive points on the convex hull of Vt, and thus the whole coloring is conflict-free.
We describe how the algorithm maintains the above invariant. A new point vt+1 that
appears at time t+1 is colored as follows: If it is inside the convex hull of Vt, then it gets color
‘?’. Otherwise, the new point vt+1will be on CH(Vt+1), in which case we essentially use the
algorithm of section 5.3 to color it. We have two case, which correspond to a substitution and
an insertion move, respectively:
• It might be the case that vt+1forces some points (say they comprise set Q) that where
in CH(Vt) to belong to the interior of CH(Vt+1), so in order to maintain the invariant, all
points in Q are recolored to ‘?’, and vt+1is colored with the maximum color occurring
in Q (this is like a substitution move of section 5.3).
• If, on the other hand, no points of CH(Vt) are forced into the convex hull, then point
vt+1∈ CH(Vt+1) is colored like in an insertion move of section 5.3, with the algorithm
for circular arcs. In that last case, in order to maintain logarithmic number of colors on
t, one recoloring of a point in CH(Vt+1) might be needed.
The total number of recolorings is guaranteed to be O(n), because for every insertion, at most
one recoloring happens on the new convex hull, and every point colored with ‘?’ remains with
that color.
6 Application to Geometry
Our randomized algorithm has applications to conflict-free colorings of certain geometric hy-
pergraphs studied in [3, 6, 10]. We obtain the same asymptotic result as in [3] but with better
constant of proportionalities and much fewer random bits.
For example, if the hypergraph H is the same as in example 7.1 it can be proved (with an
analysis similar to the one given in section 4) that for any order of insertion of n points, when
the auxiliary color for each entry is chosen uniformly at random from {a,b,c}, the expected
number of colors used is bounded by log 3
bound for dynamically coloring n points deterministically, when the whole insertion order is
known in advance, is also log 3
of colors is bounded by 1 + log8/7n ≈ 5.19log2n, whereas in our algorithm it is bounded by
1 + log3/2n ≈ 1.71log2n.
2n + 1. It is interesting that the best known upper
2n + 1 (see [2] for further details). In [3] the expected number
9
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When H is the hypergraph obtained by points in the plane intersected by halfplanes or unit
disks, we obtain online randomized algorithms that use O(logn) colors w.h.p. We summarize
it as follows:
Lemma 6.1. Let V be a finite set of n points in the plane and let E be all subsets of V that can
be obtained by intersecting V with a halfplane. Then the hypergraph H = (V,E) is 4-degenerate.
Proof. The proof uses a few geometric lemmas. Details are omitted.
Corollary 6.2. Let H be the hypergraph as in lemma 6.1. Then the expected number of colors
used by our randomized online CF-coloring applied to H is at most log 18
actual number of colors used is O(log 18
O(logn) with high probability
17n + 1. Also the
17n) with high probability. The number of random bits is
Proof. The proof follows immediately from lemma 6.1, lemma 4.4 and theorem 4.1.
Corollary 6.3. Let V be a finite set of n points in the plane and let E be all subsets of V
that can be obtained by intersecting V with a unit disc. Then there exists a randomized online
algorithm for CF-coloring H which uses O(logn) colors and O(logn) random bits with high
probability.
Proof outline. Chen, Kaplan and Sharir [10] observed that by an appropriate partitioning of the
plane one can modify any online algorithm for CF-coloring points with respect to halfplanes to
a CF-coloring of points with respect to congruent discs. Using the same technique as developed
in [10] and Corollary 6.2 we obtain the desired result.
7Discussion and Open Problems
We presented a framework for online CF-coloring any hypergraph. This framework coincides
with some known algorithms in the literature when restricted to some special underlying hy-
pergraphs. We derived a randomized online algorithm for CF-coloring any hypergraph (in the
oblivious adversary model) and showed that the performance of our algorithm depends on a
parameter which we refer to as the degeneracy of the hypergraph which is a generalization of
the known notion of degeneracy in graphs (i.e., when the hypergraph is a simple graph then
this term is the same as the classical definition of degeneracy of a graph; see Definition 2.3).
Specifically, if the hypergraph is k-degenerate then our algorithm uses O(klogn) colors w.h.p,
which is asymptotically optimal for any constant k, and O(klogklogn) random bits. This
is the first efficient online CF-coloring for general hypergraphs and subsumes all the previous
randomized algorithmic results of [3, 6, 10]. It also substantially improves the efficiency with
respect to the amount of randomness used in the special cases studied in [3, 6, 10].
Another interesting fact to note is that our algorithm when applied to k-degenerate graphs
gives an online coloring of such graphs with O(klogn) colors w.h.p. Irani [9] showed that the
same bound is achieved deterministically by the First-Fit algorithm.
It would be interesting to find an efficient online deterministic algorithm for conflict-free
coloring k-degenerate hypergraphs. Even for the very special case of a hypergraph induced by
points and intervals (as in example 7.1 where the number of neighbors of the Delaunay graph
of every induced hypergraph is at most two), the best known deterministic online CF-coloring
algorithm from [6] uses Θ(log2n) colors. We hope that our technique together with a possible
clever de-randomization technique can shed light on this problem.
10
Page 11
As mentioned already, the framework of section 3 can describe some known algorithms
such as the Unique Max Greedy of [6] for online CF-coloring points on a line with respect to
intervals. No sharp asymptotic bounds are know for the performance of Unique Max Greedy.
The best known upper bound is linear, whereas the best known lower bound is Ω(√n). We
believe that this new approach could help analyze the performance of Unique Max Greedy.
In Section 5 we initiate the study of online CF-coloring with recoloring: We provide a
deterministic online CF-coloring for points on the real line with respect to intervals and show
that our algorithm uses Θ(logn) colors and at most one recoloring per insertion. This is in
contrast with the best known deterministic online CF-coloring for this case that uses Θ(log2n)
colors in the worst case and does not recolor any other point ([6]). We also present deterministic
online algorithms for CF-coloring points with respect to circular arcs and halfplanes (and unit
discs) that use O(logn) colors and O(n) recolorings in the worst case. In the special case of
intervals or circular arcs at most one point is recolored per insertion. It would be interesting
to obtain an online CF-coloring algorithm with O(logn) and O(1) recolorings per insertion for
the case of halfplanes.
Acknowledgements
We would like to thank Amos Fiat, Haim Kaplan, J´ anos Pach, David Peleg and Micha Sharir
for helpful discussions concerning the problems studied in this paper.
References
[1] N. Alon and S. Smorodinsky. Conflict-free colorings of shallow discs. In Proc. 22nd Annual
ACM Symposium on Computational Geometry (SoCG 2006):41–43.
[2] A. Bar-Noy, P. Cheilaris, and S. Smorodinsky. Conflict-free coloring for intervals: from
offline to online. In Proc. 18th ACM Symposium on Parallelism in Algorithms and Archi-
tectures (SPAA 2006): 128–137.
[3] K. Chen. On how to play a coloring game against color-blind adversaries. In Proc. 22nd
Annual ACM Symposium on Computational Geometry (SoCG 2006): 44–51.
[4] K. Elbassioni and N. Mustafa. Conflict-Free Colorings of Rectangles Ranges. In Proc.
23rd International Symposium on Theoretical Aspects of Computer Science (STACS 2006):
254–263.
[5] G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geomet-
ric regions with applications to frequency assignment in cellular networks. SIAM Journal
on Computing, 33(1):94–136 (2003). Also in Proceedings of the 43th Annual IEEE Sym-
posium on Foundations of Computer Science (FOCS 2002).
[6] K. Chen, A. Fiat, H. Kaplan, M. Levy, J. Matouˇ sek, E. Mossel, J. Pach, M. Sharir, S.
Smorodinsky, U. Wagner, and E. Welzl. Online conflict-free coloring for intervals. SIAM
Journal on Computing, 36: 1342–1359 (2006).
[7] A. Gyarfas and J. Lehel. On-line and first fit coloring of graphs. Journal of Graph Theory,
12(2):217–227 (1988).
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[8] S. Har-Peled and S. Smorodinsky. On conflict-free coloring of points and simple regions
in the plane. Discrete and Computational Geometry, 34:47–70 (2005).
[9] S. Irani. Coloring inductive graphs on-line. Algorithmica, 11(1):53–72 (1994).
[10] K. Chen, H. Kaplan and M. Sharir. Online CF coloring for halfplanes, congruent disks,
and axis-parallel rectangles. Manuscript, 2005.
[11] H. A. Kierstead. The linearity of first-fit coloring of interval graphs. SIAM Journal on
Discrete Mathematics, 1(4):526–530 (1988).
[12] J. Pach and G. T´ oth. Conflict free colorings. In Discrete and Computational Geometry,
The Goodman-Pollack Festschrift, Springer Verlag, Heidelberg, 2003.
[13] S. Smorodinsky. Combinatorial Problems in Computational Geometry. Ph.D Dissertation,
School of Computer Science, Tel-Aviv University, 2003.
[14] S. Smorodinsky. On the chromatic number of some geometric hypergraphs. In Proceedings
of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006): 316–
323.
Appendix
An example of the framework on intervals
Example 7.1. Consider the case where the hypergraph is induced by points with respect to
intervals. Namely V = (1,...,n) and E consists of all possible discrete intervals of V (i.e.,
subsets of consecutive integers). Vertices appear one by one and at each time t we must have
an online conflict-free coloring with respect to the discrete interval subsets of the t points
revealed by time t. It is easy to see that the hypergraphs H(Vi
online 3-colored (say with auxiliary colors a, b, c): Each newly inserted point has at most two
immediate neighbors and thus even a first-fit coloring suffices. In Figure 1, we exhibit a run of
the algorithm for the permutation π = 253164, seen as a mapping from time t ∈ {1,...,6} to
the corresponding vertex at position π(t). In the end the vertices look like π−1= v4v1v3v6v2v5,
where vt is the vertex appearing at time t. The choices are f(1) = b, f(2) = a, f(3) = c,
f(4) = a, f(5) = b, f(6) = a. The six tables correspond to t = 1,...,6 and at the top of
each table the online conflict-free coloring, so far, is shown. Entries correspond to rows in the
tables, where for each entry i the following data is given: the representing color f(i) and the
proper auxiliary coloring of the vertices in the hypergraph Vi
Observe that entries 3 and 5, respectively, do not have a vertex colored with f(3) and f(5),
respectively. As a consequence colors 3, 5 do not appear in the CF-coloring although colors
1, 2, 4, 6 do. If it is important to use consecutive colors, namely k different colors implies
they are {1,...,k}, the above problem can be fixed by assigning the next unused CF-color to
an entry i only as soon as a vertex in entry i is colored with auxiliary color f(i). The above
remedy works in our general framework, not only in the specific case of this example.
t) can always be properly
twith three colors a,b or c.
Proof of lemma 3.1
Lemma. If the framework halts for any vertex vtthen it produces a valid online CF-coloring
of H.
12
Page 13
i
1
2
3
4
5
6
f(i)
b
a
· 2 · · · ·
a
a
i
1
2
3
4
5
6
f(i)
b
a
· 2 · · 1 ·
a
a
b
i
1
2
3
4
5
6
f(i)
b
a
c
a
· 2 4 · 1 ·
a c
a b
a
a
b
i
1
2
3
4
5
6
f(i)
b
a
c
a
1 2 4 · 1 ·
b a c
a b
a
a
b
i
1
2
3
4
5
6
f(i)
b
a
c
a
1 2 4 · 1 2
b a c
a b
a
a
b a
a
i
1
2
3
4
5
6
f(i)
b
a
c
a
b
a
1 2 4 6 1 2
b a c a b a
a b c
a b
a b
a
a
a
Figure 1: A run example of the framework for hypergraphs induced by points with respect to
intervals. In this case the hypergraph admits a proper online 3-coloring in any entry of the
table. The auxiliary colors are denoted by a,b,c.
Proof. Let H(Vt) be the hypergraph induced by the vertices revealed at time t. Let S be a
hyperedge in this hypergraph and let j be the maximum integer for which there is a vertex
v of S colored with j. We claim that exactly one such vertex in S exists. Assume to the
contrary that there is another vertex v?in S colored with j. This means that at time t both
vertices v and v?were present at entry j of the table (i.e., v,v?∈ Vj
an auxiliary color (in the auxiliary coloring of the set Vj
the auxiliary coloring is a proper coloring of the induced hypergraph at entry j, S ∩ Vj
monochromatic so there must exists a third vertex v??∈ S ∩ Vj
and was assigned an auxiliary color different from f(j). Thus v??got its final color in an entry
greater than j, a contradiction to the maximality of j in the hyperedge S. This completes the
proof of the lemma.
t) and that they both got
t) which equals f(j). However, since
tis not
tthat was present at entry j
Proof of lemma 4.2
Lemma. Let H = (V,E) be a k-degenerate hypergraph and let Vj
t and at level j as produced by the above algorithm. Then for any j and t if vtis assigned a
color distinct from all its neighbors in the Delaunay graph G(H(Vj
with the colors assigned to the vertices Vj
tbe the subset of V at time
t)) then this color together
t−1is also a proper coloring of the hypergraph H(Vj
t−1) is properly colored by the
t).
Proof. By induction on t. The induction hypothesis is that H(Vj
auxiliary coloring. Let vtbe the vertex added to the hypergraph induced by the j’th entry at
time t. Any hyperedge S that contains at least two vertices of Vj
not monochromatic by the induction hypothesis. Thus, we are only concerned with hyperedges
of cardinality two that contain vtand exactly one vertex of Vj
vt got a color that is distinct from any vertex u such that {u,vt} is a hyperedge of H(Vj
(Those are exactly the neighbors of vtin the corresponding Delaunay graph). Thus any such
hyperedge {u,vt} is also not monochromatic. This completes the inductive step and hence the
proof of the lemma.
t−1or does not contain vtis
t−1. However, we assumed that
t)
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Page 14
An example of a run of the recoloring algorithm for intervals
See figure 2.
· · v1 · · · v2
a1
2
3
d
a
· · 1 · · · 2
· · v1 · v3 · v2
a1
2
3
d
a
a
· · 1 · 2 · 1
· · v1v4v3 · v2
a d d
d a
a
· · 1 3 2 · 1
1
2
3
a
· · v1v4v3v5v2
a d a d a
d
a
· · 1 3 1 2 1
1
2
3
a
v6 · v1v4v3v5v2
d a d a d a
a
d
a
2 · 1 3 1 2 1
1
2
3
a
v6v7v1v4v3v5v2
a d a d a d a
ad
a
1 2 1 3 1 2 1
1
2
3
a
Figure 2: An example run of the recoloring algorithm. The first row of the table represents
the order in which points appeared, the last row of the table shows current color allocation.
The first table is at t = 2, after the first two points have appeared. At every time step of the
run, points whose colors were changed (a new color, or a recoloring) by the last insertion are
marked with bold.
14
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