We consider the well-known vertex coloring problem: given a graph G, find a coloring of the vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree ∆ can be colored with ∆+1 colors, and distributed algorithms that find a (∆+1)-coloring in a logarithmic number of communication rounds, with high probability, are known since more than a decade. This is in general the best possible if only a constant number of bits can be sent along every edge in each round. In fact, we show that for the n-node cycle the bit complexity of the coloring problem is Ω(log n). More precisely, if only one bit can be sent along each edge in a round, then every distributed coloring algorithm (i.e., algorithms in which every node has the same initial state and initially only knows its own edges) needs at least Ω(log n) rounds, with high probability, to color the cycle, for any finite number of colors. But what if the edges have orientations, i.e., the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Does this allow one to provide faster coloring algorithms?Interestingly, for the cycle in which all edges have the same orientation, we show that a simple randomized algorithm can achieve a 3-coloring with only O( √ log n) rounds of bit transmissions, with high probability (w.h.p.). Large Scale Information Systems (DELIS).sult is tight because we also show that the bit complexity of coloring an oriented cycle is Ω( √ log n), with high probability, no matter how many colors are allowed. The 3-coloring algorithm can be easily extended to provide a (∆ + 1)-coloring for all graphs of maximum degree ∆ in O( √ log n) rounds of bit transmissions, w.h.p., if ∆ is a constant, the edges are oriented, and the graph does not contain an oriented cycle of length less than √ log n. Using more complex algorithms, we show how to obtain an O(∆)-coloring for arbitrary oriented graphs of maximum degree ∆ using essentially O(log ∆ + √ log n) rounds of bit transmissions, w.h.p., provided that the graph does not contain an oriented cycle of length less than √ log n.
An important problem for wireless ad hoc networks has been to design overlay networks that allow time-and energy-efficient routing. Many local-control strategies for maintaining such overlay networks have already been suggested, but most of them are based on an oversimplified wireless communication model.In this paper, we suggest a model that is much more general than previous models. It allows the path loss of transmissions to significantly deviate from the idealistic unit disk model and does not even require the path loss to form a metric. Also, our model is apparently the first proposed for algorithm design that does not only model transmission and interference issues but also aims at providing a realistic model for physical carrier sensing. Physical carrier sensing is needed so that our protocols do not require any prior information (not even an estimate on the number of nodes) about the wireless network to run efficiently.Based on this model, we propose a local-control protocol for establishing a constant density spanner among a set of mobile stations (or nodes) that are distributed in an arbitrary way in a 2-dimensional Euclidean space. More precisely, we establish a backbone structure by efficiently electing cluster leaders and gateway nodes so that there is only a constant number of cluster leaders and gateway nodes within the transmission range of any node and the backbone structure satisfies the properties of a topological spanner.Our protocol has the advantage that it is locally self-stabilizing, i.e., it can recover from any initial configuration, even if adversarial nodes participate in it, as long as the honest nodes sufficiently far away from adversarial nodes can in principle form a single connected component. Furthermore, we only need constant size messages and a constant amount of storage at the nodes, irrespective of the distribution of the nodes. Hence, our protocols would even work in extreme situations such as very simple wireless devices (like sensors) in a hostile environment.
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