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"... Abstract. In the paper we present a distributed probabilistic algorithm for coloring the vertices of a graph. Since this algorithm resembles a largest-first strategy, we call it the distributed LF (DLF) algorithm. The coloring obtained by DLF is optimal or near optimal for numerous classes of graphs ..."

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Abstract. In the paper we present a distributed probabilistic algorithm for coloring the vertices of a graph. Since this algorithm resembles a largest-first strategy, we call it the distributed LF (DLF) algorithm. The coloring obtained by DLF is optimal or near optimal for numerous classes of graphs e.g. complete k-partite, caterpillars, crowns, bipartite wheels. We also show that DLF runs in O(\Delta 2 log n) rounds for an arbitrary graph, where n is the number of vertices and \Delta denotes the largest vertex degree. 1 Introduction We discuss the vertex coloring problem in a distributed network. Such a network consists of processors and bidirectional communication links between pairs of them. It can be modeled by a graph G = (V; E). The set of vertices V corresponds to processors and the set E of edges models links in the network. To color the vertices of G means to give each vertex a color in such a way that no two adjacent vertices get the same color. If at most k colors are used, the result is called a k-coloring. We assume that there is no shared memory. Each processor knows its own links and its unique identifier. We want these units to compute a coloring of the associated graph without any other information about the structure of G. We assume that the system is synchronized in rounds. The number of rounds will be our measure of the time complexity. Such a model of coloring can be used in a distributed multihop wireless network to eliminate packet collisions by assigning orthogonal codes to radio stations [1].

### Rendezvous of Heterogeneous Mobile Agents in Edge-weighted Networks?

"... Abstract. We introduce a variant of the deterministic rendezvous prob-lem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial position ..."

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Abstract. We introduce a variant of the deterministic rendezvous prob-lem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the correspond-ing traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time TOPT in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time O(nTOPT) in a n-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to Θ(TOPT) when the agents are allowed to exchange Θ(n) bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time Θ(TOPT). 1

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