## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, May 08, 2007, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Fabian Kuhn

## On the Complexity of Distributed Graph Coloring

Coloring the nodes of a graph with a small number of colors is one of the most fundamental problems in theoretical computer science. We study graph coloring in a distributed setting. Processors of a distributed system are nodes of an undirected graph G. There is an edge between two nodes whenever the corresponding processors can directly communicate with each other. We assume that distributed coloring algorithms start with an initial m-coloring of G. We prove new lower bounds for two special kinds of coloring algorithms. For algorithms which run for a single communication round---i.e., every node of the network can only send its initial color to all its neighbors---, we show that the number of colors of the computed coloring has to be at least Omega(Delta^2/log^2(Delta)+loglog m) where Delta is the largest degree of G. It is known that it is possible to color G with O(Delta^2*log m) colors in one round. If such one-round algorithms are iteratively applied to reduce the number of colors step-by-step, we prove a time lower bound of Omega(Delta/log^2(Delta)+log^*m) to obtain an O(Delta)-coloring where the best known upper bound are O(Delta*log(Delta)+log^*m) rounds. The best previous lower bounds for the two types of algorithms are Omega(loglog m) colors and Omega(log^*m) rounds, respectively.

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