Deterministic Dispersion of Mobile Robots in Dynamic Rings

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.~\cite{AM17}, 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 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.~\cite{LDFS16}, 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.