Title: Deterministic Dispersion of Mobile Robots in Dynamic Rings,
Abstract: In this work, we study the problem of dispersion of mobile robots on dynamic
rings. The problem of dispersion of $n$ robots on an $n$ node graph, introduced
by Augustine and Moses Jr. [1], requires robots to coordinate with each other
and reach a configuration where exactly one robot is present on each node. This
problem has real world applications and applies whenever we want to minimize
the total cost of $n$ agents sharing $n$ resources, located at various places,
subject to the constraint that the cost of an agent moving to a different
resource is comparatively much smaller than the cost of multiple agents sharing
a resource (e.g. smart electric cars sharing recharge stations). The study of
this problem also provides indirect benefits to the study of scattering on
graphs, the study of exploration by mobile robots, and the study of load
balancing on graphs.
We solve the problem of dispersion in the presence of two types of dynamism
in the underlying graph: (i) vertex permutation and (ii) 1-interval
connectivity. We introduce the notion of vertex permutation dynamism and have
it mean that for a given set of nodes, in every round, the adversary ensures a
ring structure is maintained, but the connections between the nodes may change.
We use the idea of 1-interval connectivity from Di Luna et al. [10], where for
a given ring, in each round, the adversary chooses at most one edge to remove.
We assume robots have full visibility and present asymptotically time optimal
algorithms to achieve dispersion in the presence of both types of dynamism when
robots have chirality. When robots do not have chirality, we present
asymptotically time optimal algorithms to achieve dispersion subject to certain
constraints. Finally, we provide impossibility results for dispersion when
robots have no visibility.