The gathering over meeting nodes problem asks the robots to gather at one of
the pre-defined meeting nodes. The robots are deployed on the nodes of an
anonymous two-dimensional infinite grid which has a subset of nodes marked as
meeting nodes. Robots are identical, autonomous, anonymous and oblivious. They
operate under an asynchronous scheduler. They do not have any agreement on a
global coordinate system. All the initial configurations for which the problem
is deterministically unsolvable have been characterized. A deterministic
distributed algorithm has been proposed to solve the problem for the remaining
configurations. The efficiency of the proposed algorithm is studied in terms of
the number of moves required for gathering. A lower bound concerning the total
number of moves required to solve the gathering problem has been derived.
A number-conserving cellular automaton is a simplified model for a system of
interacting particles. This paper contains two related constructions by which
one can find all one-dimensional number-conserving cellular automata with one
kind of particle.
The output of both methods is a "flow function", which describes the movement
of the particles. In the first method, one puts increasingly stronger
restrictions on the particle flow until a single flow function is specified.
There are no dead ends, every choice of restriction steps ends with a flow.
The second method uses the fact that the flow functions can be ordered and
then form a lattice. This method consists of a recipe for the slowest flow that
enforces a given minimal particle speed in one given neighbourhood. All other
flow functions are then maxima of sets of these flows.
Other questions, like that about the nature of non-deterministic
number-conserving rules, are treated briefly at the end.
In communication networks, the binding numbers of graphs (or networks) are
often used to measure the vulnerability and robustness of graphs (or networks).
Furthermore, the fractional factors of graphs and the fractional
ID-$[a,b]$-factor-critical covered graphs have a great deal of important
applications in the data transmission networks. In this paper, we investigate
the relationship between the binding numbers of graphs and the fractional
ID-$[a,b]$-factor-critical covered graphs, and derive a binding number
condition for a graph to be fractional ID-$[a,b]$-factor-critical covered,
which is an extension of Zhou's previous result [S. Zhou, Binding numbers for
fractional ID-$k$-factor-critical graphs, Acta Mathematica Sinica, English
Series 30(1)(2014)181--186].