Distributed soft path coloring

Paper in proceeding, 2003

Soft coloring of a graph means that only a few adjacent vertices receive the same color. We study soft coloring in the distributed model where vertices are processing units and edges are communication links. We aim at reducing coloring conflicts as quickly as possible over time by recoloring. Basic insights can be obtained already on the simplest topology, a path. We propose a randomized algorithm for 2-coloring the path with optimal decrease of conflicts. Its analysis is based on properties of random strings of brackets. Conflicts can be reduced exponentially faster if extra colors are allowed. Thus we observe the existence of two significantly different types of the problem, with different ways to get rid of conflicts. Later we generalize the results to a broader class of locally checkable labeling problems on enhanced paths, namely labelings by states from a skew-symmetric transition graph. A result for grid coloring is also presented.