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13
Online ConflictFree Coloring for Intervals
, 2006
"... We consider an online version of the conflictfree coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflictfree, in the sense that in every interval I there is a color that appears exactly once ..."
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Cited by 26 (6 self)
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We consider an online version of the conflictfree coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflictfree, in the sense that in every interval I there is a color that appears exactly once in I. We present deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω ( √ n) colors in the worst case. We then derive two efficient variants of this simple algorithm. The first is deterministic and uses O(log 2 n) colors, and the second is randomized and uses O(log n) colors with high probability. We also show that the O(log 2 n) bound on the number of colors used by our deterministic algorithm is tight on the worst case. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high
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 25 (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).
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 11 (2 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 8 (2 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 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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Cited by 5 (0 self)
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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
Online conflict free coloring for halfplanes, congruent disks, and axisparallel rectangles
, 2008
"... 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(log³ n) colors. This is the first efficient (that is, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.
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