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13
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 of Points with Respect to Rectangles and Approximation Algorithms for Discrete Independent Set
, 2012
"... In the conflictfree coloring problem, for a given range space, we want to bound the minimum value F (n) such that every set P of n points can be colored with F (n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n0.368) with respect to orthog ..."
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In the conflictfree coloring problem, for a given range space, we want to bound the minimum value F (n) such that every set P of n points can be colored with F (n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n0.368) with respect to orthogonal ranges in two dimensions (i.e., axisparallel rectangles), which is the first improvement over the previous bound O(n0.382) by Ajwani, Elbassioni, Govindarajan, and Ray [SPAA’07]. This result leads to an O(n1−0.632/2d−2) upper bound with respect to orthogonal ranges (boxes) in dimension d, and also an O(n1−0.632/(2d−3−0.368) ) upper bound with respect to dominance ranges (orthants) in dimension d ≥ 4. We also observe that combinatorial results on conflictfree coloring can be applied to the analysis of approximation algorithms for discrete versions of geometric independent set problems. Here, given a set P of (weighted) points and a set S of ranges, we want to select the largest(weight) subset Q ⊂ P with the property that every range of S contains at most one point of Q. We obtain, for example, a randomized O(n0.368)approximation algorithm for this problem with respect to orthogonal ranges in the plane. 1
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.
Online conflict free coloring for halfplanes, congruent disks, and axisparallel rectangles. Manuscript
"... We present randomized algorithms for online conflictfree coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearlyequal axisparallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability. We also present a d ..."
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We present randomized algorithms for online conflictfree coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearlyequal axisparallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability. We also present a deterministic algorithm for online CF coloring of points in the plane with respect to nearlyequal axisparallel rectangles, using O(log 3 n) colors. This is the first efficient (that is, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.