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Distributed Coloring with O˜(√log n) Bits
K Kothapalli, M Onus, C Scheideler, C Schindelhauer
Proc. of IEEE International Parallel and Distributed Processing Symposium …
We consider the well-known vertex coloring problem: given a graph G, find a coloring of its 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 (ie, 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 n–node cycle, for any finite number of colors. But what if the edges have orientations, ie, the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Edge orientations naturally occur in dynamic networks where new nodes establish connections to old nodes. Does this allow one to provide faster coloring algorithms?