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10
Monadic Secondorder Logic for Parameterized Verification
 in: Proc. 19th Symp. on Parallelism in Algorithms and Architectures (SPAA
, 1994
"... Given a set of points P ⊆ R 2, a conflictfree coloring of P is an assignment of colors to points of P, such that each nonempty axisparallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is ..."
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Cited by 13 (0 self)
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Given a set of points P ⊆ R 2, a conflictfree coloring of P is an assignment of colors to points of P, such that each nonempty axisparallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflictfree colored with Õ(n.382+ɛ) colors in expected polynomial time, for any arbitrarily small ɛ> 0. This improves upon the previously known bound of O ( p n log log n/log n).
ConflictFree Coloring for Intervals: from Offline to Online (Extended Abstract)
, 2006
"... This paper studies deterministic algorithms for a frequency assignment problem in cellular networks. A cellular network consists of fixedposition base stations and moving agents. Each base station operates at a fixed frequency, and this allows an agent tuned at this frequency to communicate with th ..."
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Cited by 11 (5 self)
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This paper studies deterministic algorithms for a frequency assignment problem in cellular networks. A cellular network consists of fixedposition base stations and moving agents. Each base station operates at a fixed frequency, and this allows an agent tuned at this frequency to communicate with the base station. Each agent has a specific range of communication (described as a geometric shape, e.g., a disc) that may contain one or several base stations. To avoid interference, the goal is to assign frequencies to base stations such that for any range, there exists a base station in the range with a frequency that is not reused by some other base station in the range. The base station with this unique (in the range) frequency serves the aforementioned range. Since using many frequencies is expensive, the optimization goal is to use as few frequencies as possible. The problem can be modeled as a special coloring problem for hypergraphs. Base stations are the vertices, ranges are the hyperedges, and colors (frequencies) must be assigned to vertices following the conflictfree property: In every hyperedge there is a color that occurs exactly once. We concentrate on the special case where the n base stations lie on the real line and ranges are the n(n + 1)/2 nonempty subsets of consecutive points. This problem is called conflictfree coloring for intervals. We introduce a hierarchy of four models for the above problem: (i) the static model, where the complete hypergraph is given and all vertices are colored simultaneously, (ii) the dynamic offline model, where the vertices appear in some order and the conflictfree prop
How to play a coloring game against a colorblind adversary
 In Proc. 22nd Annual ACM Symposium on Computational Geometry (SoCG 2006
, 2006
"... We study the problem of conflictfree (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearlyequal axisparallel rectangles, and congruent disks. As a warmup exercise, the online CF coloring of points on the line with respect to intervals is also con ..."
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Cited by 11 (3 self)
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We study the problem of conflictfree (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearlyequal axisparallel rectangles, and congruent disks. As a warmup exercise, the online CF coloring of points on the line with respect to intervals is also considered. We present randomized algorithms in the oblivious adversary model, where the adversary does not see the colors used. For the problems considered, the algorithms always produce valid CF colorings, and use O(log n) colors with high probability (these bounds are optimal in the worst case). Our randomized online algorithms are considerably simpler than previous algorithms for this problem and use fewer colors. We also present a deterministic algorithm for the CF coloring of points in the plane with respect to nearlyequal axisparallel rectangles, using O(polylog(n)) colors. This is the first efficient deterministic online CF coloring algorithm for this problem. 1
Online Conflictfree Colorings for Hypergraphs
, 2007
"... We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically ..."
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Cited by 8 (3 self)
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We provide a framework for online conflictfree coloring (CFcoloring) of any hypergraph. We use this framework to obtain an efficient randomized online algorithm for CFcoloring any kdegenerate hypergraph. Our algorithm uses O(k log n) colors with high probability and this bound is asymptotically optimal for any constant k. Moreover, our algorithm uses O(k log k log n) random bits with high probability. As a corollary, we obtain asymptotically optimal randomized algorithms for online CFcoloring some hypergraphs that arise in geometry. Our algorithm uses exponentially fewer random bits compared to previous results. We introduce deterministic online CFcoloring 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(log n) colors and recolor O(n) points in total.
ConflictFree Coloring and its Applications
, 2010
"... Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to c ..."
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Cited by 2 (0 self)
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Let H = (V, E) be a hypergraph. A conflictfree coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols and several other fields, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.
Conflictfree colorings of graphs and hypergraphs
"... A coloring of the vertices of a hypergraph H is called conflictfree if each hyperedge E of H contains a vertex of “unique ” color that does not get repeated in E. The smallest number of colors required for such a coloring is called the conflictfree chromatic number of H, and is denoted by χCF(H). ..."
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Cited by 1 (0 self)
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A coloring of the vertices of a hypergraph H is called conflictfree if each hyperedge E of H contains a vertex of “unique ” color that does not get repeated in E. The smallest number of colors required for such a coloring is called the conflictfree chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even et al. (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyze this notion for general hypergraphs. It is shown that χCF(H) ≤ 1/2 + √ 2m + 1/4, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m 1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász’s Local Lemma, the same result holds for hypergraphs, in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial time algorithms to obtain such colorings. Our machinery can also be applied to the hypergraphs induced by the neighborhoods of the vertices of a graph. It turns out that in this case we need much fewer colors. For example, it is shown that the vertices of any graph G with maximum degree ∆ can be colored with log 2+ǫ ∆ colors, so that the neighborhood of every vertex contains a point of “unique ” color. We give an efficient deterministic algorithm to find such a coloring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need (1) to correct a small error in the MolloyReed approach; (2) to restate and reprove their result in a deterministic form.
Dynamic Offline ConflictFree Coloring for Unit Disks
"... Abstract. A conflictfree coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color different from that of the rest of the disks that covers p. In the dynamic offline setting, a sequence of disks is g ..."
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Abstract. A conflictfree coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color different from that of the rest of the disks that covers p. In the dynamic offline setting, a sequence of disks is given, we have to color the disks onebyone according to the order of the sequence and maintain the conflictfree property at any time for the disks that are colored. This paper focuses on unit disks, i.e., disks with radius one. We give an algorithm that colors a sequence of n unit disks in the dynamic offline setting using O(log n) colors. The algorithm is asymptotically optimal because Ω(log n) colors is necessary to color some set of n unit disks for any value of n [9]. 1
ConflictFree Coloring of Points on a Line with respect to a Set of Intervals
"... We present a 2approximation algorithm for CFcoloring of points on a line with respect to a given set of intervals. The running time of the algorithm is O(nlog n). 1 ..."
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We present a 2approximation algorithm for CFcoloring of points on a line with respect to a given set of intervals. The running time of the algorithm is O(nlog n). 1
ON THE CHROMATIC NUMBER OF GEOMETRIC HYPERGRAPHS
 VOL. 21, NO. 3, PP. 676–687
, 2007
"... A finite family R of simple Jordan regions in the plane defines a hypergraph H = H(R) where the vertex set of H is R and the hyperedges are all subsets S ⊂Rfor which there is a point p such that S = {r ∈Rp ∈ r}. The chromatic number of H(R) is the minimum number of colors needed to color the membe ..."
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A finite family R of simple Jordan regions in the plane defines a hypergraph H = H(R) where the vertex set of H is R and the hyperedges are all subsets S ⊂Rfor which there is a point p such that S = {r ∈Rp ∈ r}. The chromatic number of H(R) is the minimum number of colors needed to color the members of R such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs and obtain the following results: (i) Any hypergraph that is induced by a family of n simple Jordan regions such that the maximum union complexity of any k of them (for 1 ≤ k ≤ m) is bounded by U(m) and U(m) m is a nondecreasing function is O ( U(n))colorable. Thus, for example, we prove that any finite family of pseudodiscs can n be colored with a constant number of colors. (ii) Any hypergraph induced by a finite family of planar discs is four colorable. This bound is tight. In fact, we prove that this statement is equivalent to the fourcolor theorem. (iii) Any hypergraph induced by n axisparallel rectangles is O(log n)colorable. This bound is asymptotically tight. Our proofs are constructive. Namely, we provide deterministic polynomialtime algorithms for coloring such hypergraphs with only “few ” colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs). As an application of (i) and (ii) we obtain simple constructive proofs for the following: (iv) Any set of n Jordan regions with near linear union complexity admits a conflictfree (CF) coloring with polylogarithmic number of colors. (v) Any set of n axisparallel rectangles admits a CFcoloring with O(log2 (n)) colors.
A Lower Bound on the Area of a 3Coloured Disk Packing ∗
, 2008
"... Given a set of unitdisks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved 1/4.41. Motivated by the problem of channelassignment for wireless access points, in which use of 3 channels is a standar ..."
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Given a set of unitdisks in the plane with union area A, what fraction of A can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved 1/4.41. Motivated by the problem of channelassignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3colourable with disks of the same colour pairwisedisjoint. For this variant of the problem, we conjecture that it is always possible to cover at least 1/1.41 of the union area and prove 1/2.09. We also provide an O(n 2) algorithm to select a subset achieving a 1/2.77 bound. 1