  This article is on message-passing systems where communication is (a)
synchronous and (b) based on the "broadcast/receive" pair of communication
operations. "Synchronous" means that time is discrete and appears as a sequence
of time slots (or rounds) such that each message is received in the very same
round in which it is sent. "Broadcast/receive" means that during a round a
process can either broadcast a message to its neighbors or receive a message
from one of them. In such a communication model, no two neighbors of the same
process, nor a process and any of its neighbors, must be allowed to broadcast
during the same time slot (thereby preventing message collisions in the first
case, and message conflicts in the second case). From a graph theory point of
view, the allocation of slots to processes is know as the distance-2 coloring
problem: a color must be associated with each process (defining the time slots
in which it will be allowed to broadcast) in such a way that any two processes
at distance at most 2 obtain different colors, while the total number of colors
is "as small as possible". The paper presents a parallel message-passing
distance-2 coloring algorithm suited to trees, whose roots are dynamically
defined. This algorithm, which is itself collision-free and conflict-free, uses
$\Delta + 1$ colors where $\Delta$ is the maximal degree of the graph (hence
the algorithm is color-optimal). It does not require all processes to have
different initial identities, and its time complexity is $O(d \Delta)$, where d
is the depth of the tree. As far as we know, this is the first distributed
distance-2 coloring algorithm designed for the broadcast/receive round-based
communication model, which owns all the previous properties.

Distributed, Parallel, and Cluster Computing;Networking and Internet Architecture