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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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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
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.
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
CONFLICTFREE COLORINGS OF UNIFORM HYPERGRAPHS WITH FEW EDGES
, 2012
"... 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 prov ..."
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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.
Kinetic ConflictFree Coloring∗
"... A conflictfree coloring, or CFcoloring for short, of a set P of points in the plane with respect to disks is a coloring of the points of P with the following property: for any disk D containing at least one point of P there is a point p ∈ P ∩D so that no other point q ∈ P ∩D has the same color as ..."
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A conflictfree coloring, or CFcoloring for short, of a set P of points in the plane with respect to disks is a coloring of the points of P with the following property: for any disk D containing at least one point of P there is a point p ∈ P ∩D so that no other point q ∈ P ∩D has the same color as p. In this paper we study the problem of maintaining such a CFcoloring when the points in P move. We present two methods for this and evaluate the maximum number of colors used as well as the number of recolorings, both in theory and experimentally. 1
Pathrelated vertex colorings of graphs
"... We investigate algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special coloring problem for graphs. Base stations are the vertices, ranges are the paths in the graph, and colors (frequencies) must be assigned to vertices following the conflictfree ..."
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We investigate algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special coloring problem for graphs. Base stations are the vertices, ranges are the paths in the graph, and colors (frequencies) must be assigned to vertices following the conflictfree property: In every path there is a color that occurs exactly once. We concentrate on the special case where the base stations lie on a chain and ranges are the nonempty subchains. We also consider other simple graphs, such as rings, trees, and grids. We discuss a whole hierarchy of related coloring problems.