Conference Paper

# Online Conflict-Free Colorings for Hypergraphs.

DOI: 10.1007/978-3-540-73420-8_21 Conference: Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings

Source: DBLP

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**ABSTRACT:**A con∞ict-free 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 difierent from that of the rest of the disks that covers p. In the dynamic o†ine setting, a sequence of disks is given, we have to color the disks one-by-one according to the order of the sequence and maintain the con∞ict-free 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 o†ine setting using O(logn) colors. The algorithm is asymptotically optimal because ›(logn) colors is necessary to color some set of n unit disks for any value of n (8). - [Show abstract] [Hide abstract]

**ABSTRACT:**In a coloring of a set of points P with respect to a family of geometric regions one requires that in every region containing at least two points from P, not all the points are of the same color. Perhaps the most notorious open case is coloring of n points in the plane with respect to axis-parallel rectangles, for which it is known that O(n0.368)O(n0.368) colors always suffice, and Ω(logn/log2logn) colors are sometimes necessary.In this note we give a simple proof showing that every set P of n points in the plane can be colored with O(logn) colors such that every axis-parallel rectangle that contains at least three points from P is non-monochromatic.Journal of Combinatorial Theory Series A 05/2013; 120(4):811–815. · 0.87 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Some of the routing protocols used in telecommunication networks route traffic on a shortest path tree according to configurable integral link weights. One crucial issue for network operators is finding a weight function that ensures a stable routing: when some link fails, traffic whose path does not use that link should not be rerouted. In this paper we improve on several previously best results for finding small stable weights. As a conceptual contribution, we draw a connection between the stable weights problem and the seemingly unrelated unique-max coloring problem. In unique-max coloring, one is given a set of points and a family of subsets of those points called regions. The task is to assign to each region a color represented as an integer such that, for every point, one region containing it has a color strictly larger than the color of any other region containing this point. In our setting, points and regions become edges and paths of the shortest path tree, respectively, and based on this connection, we provide stable weight functions with a maximum weight of O(nlogn) in the case of single link failure, where n is the number of vertices in the network. Furthermore, if the root of the shortest path tree is known, we present an algorithm for determining stable weights bounded by 4n, which is optimal up to constant factors. For the case of an arbitrary number of failures, we show how stable weights bounded by 3 n n can be obtained. All the results improve on the previously best known bounds.SIAM Journal on Discrete Mathematics 01/2013; 27(1). · 0.58 Impact Factor

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