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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.
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
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"... Delaunay graphs of point sets in the plane with respect to axisparallel rectangles ..."
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Delaunay graphs of point sets in the plane with respect to axisparallel rectangles
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"... Delaunay graphs of point sets in the plane with respect to axisparallel rectangles ..."
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Delaunay graphs of point sets in the plane with respect to axisparallel rectangles
BenGurion University
"... We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflictfree coloring, in every path ..."
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We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflictfree colorings. In a uniquemaximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflictfree coloring, in every path of the graph there is a color that appears only once. We also study computational complexity aspects of conflictfree colorings and prove a completeness result. Finally, we improve lower bounds for those chromatic numbers of the grid graph.
Ordered coloring of grids and related graphsI
"... 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. Key words: grid graph, ordered coloring, vertex ranking, conflictfree coloring 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.
Deterministic conflictfree . . .
"... We investigate deterministic algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special vertex coloring problem for hypergraphs: In every hyperedge there must exist a vertex with a color that occurs exactly once in the hyperedge (the conflictfree pro ..."
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We investigate deterministic algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special vertex coloring problem for hypergraphs: In every hyperedge there must exist a vertex with a color that occurs exactly once in the hyperedge (the conflictfree property). We concentrate on a special case of the problem, called conflictfree coloring for intervals. We introduce a hierarchy of four models for the above problem: (i) static, (ii) dynamic offline, (iii) dynamic online with absolute positions, (iv) dynamic online with relative positions. In the dynamic offline model, we give a deterministic algorithm that uses at most log3/2 n + 1 ≈ 1.71 log2 n colors and exhibit inputs that force any algorithm to use at least 3 log5 n+ 1 ≈ 1.29 log2 n colors. For the online absolute positions model, we give a deterministic algorithm that uses at most 3dlog3 ne ≈ 1.89 log2 n colors. To the best of our knowledge, this is the first deterministic online algorithm using O(logn) colors, in a nontrivial online model. In the online relative positions model, we resolve an open problem by showing a tight analysis on the number of colors used by the firstfit greedy online algorithm. We also consider conflictfree coloring only with respect to intervals that contain at least one of the two extreme points.