We investigate the class of so-called

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.

We present k-Flipper, a graph transformation algorithm that transforms regular undirected graphs. Given a path of k + 2 edges it interchanges the end vertices of the path. By definition this operation preserves regularity and connectivity. We show that every regular connected graph can be reached by a series of these operations for all k ≥ 1. We use a randomized version, called Random k-Flipper, in order to create random regular connected undirected graphs that may serve as a backbone for peer-to-peer networks. We prove for degree d ∈ Ω(log n) that a series of O(dn) Random k-Flipper operations with k ∈ Θ(d 2 n 2 log 1/ ) transforms any graph into an expander graph with high probability, i.e. 1−n −Θ(1) . The Random 1-Flipper is symmetric, i.e. the transformation probability from any labeled d-regular graph G to G is equal to those from G to G. From this and the reachability property we conclude that in the limit a series of Random 1-Flipper operations converges against an uniform probability distribution over all connected labeled d-regular graphs. For degree d ∈ ω(1) growing with the graph size this implies that iteratively applying Random 1-Flipper transforms any given graph into an expander asymptotically almost surely.We use these operations as a maintenance operation for a peer-to-peer network based on random regular connected graphs that provides high robustness and recovers from degenerate network structures by continuously applying these random graph transformations. For this, we describe how network operations for joining and leaving the network can be designed and how the concurrency of the graph transformations can be handled.

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