Locality in Distributed Graph Algorithms - Linial, Nathan (SIAM journal on computing, 1992)
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 $2r/3$ requires at least $Ω (\sqrt d )$ colors. * In an $n$-vertex graph of largest degree $Δ $, an $O(Δ ^2 )$-coloring may be found in time $O(\log ^ * n)$.