Locality in distributed graph algorithms

This paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion. The computational model is such that each node of the graph is occupied by a processor which has its own ID. Processors are restricted to collecting data from others which are at a distance at most t away from them in t time units, but are otherwise computationally unbounded. This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data. Three results are proved within this model. A 3-coloring of an n-cycle requires time Ω(log n). This bound is tight, by previous work of Cole and Vishkin. Any algorithm for coloring the d-regular tree of radius r which runs for time at most 2τ/3 requires at least Ω(√d) colors. In an n-vortex graph of largest degree Δ, O(Δ²)-coloring may be found in time O(log n).