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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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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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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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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 COLORINGS OF UNIFORM HYPERGRAPHS WITH FEW EDGES
"... Abstract. A coloring of the vertices of a hypergraph H is called conflictfree if each edge e of H contains a vertex whose color does not repeat 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). Pach and Tar ..."
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Abstract. A coloring of the vertices of a hypergraph H is called conflictfree if each edge e of H contains a vertex whose color does not repeat 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). Pach and Tardos proved that for an (2r − 1)uniform hypergraph H with m edges, χCF (H) is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for runiform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an runiform simple hypergraph that is not conflictfree kcolorable. 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.
Ordered coloring grids and related graphs
"... We investigate a coloring problem, called ordered coloring, in grids and some other families of gridlike graphs. Ordered coloring (also known as vertex ranking) is related to conflictfree coloring and other traditional coloring problems. Such coloring problems can model (among others) efficient fr ..."
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We investigate a coloring problem, called ordered coloring, in grids and some other families of gridlike graphs. Ordered coloring (also known as vertex ranking) is related to conflictfree coloring and other traditional coloring problems. Such coloring problems can model (among others) efficient frequency assignments in cellular networks. Our main technical results improve upper and lower bounds for the ordered chromatic number of grids and related graphs. To the best of our knowledge, this is the first attempt to calculate exactly the ordered chromatic number of these graph families.