Abstract:
The problem of dispersion of mobile robots on a graph asks
that n robots initially placed arbitrarily on the nodes of an n-node anonymous graph, autonomously move to reach a final configuration where exactly each node has at most one robot on it. This problem is of significant
interest due to its relationship to other fundamental robot coordination
problems, such as exploration, scattering, load balancing, relocation of
self-driving electric cars to recharge stations, etc. The robots have unique
IDs, typically in the range [1, poly(n)] and limited memory, whereas the
graph is anonymous, i.e., the nodes do not have identifiers. The objective is to simultaneously minimize two performance metrics: (i) time
to achieve dispersion and (ii) memory requirement at each robot. This
problem has been relatively well-studied when robots are non-faulty.
In this paper, we introduce the notion of Byzantine faults to this problem, i.e., we formalize the problem of dispersion in the presence of up to
f Byzantine robots. We then study the problem on a ring while simultaneously optimizing the time complexity of algorithms and the memory
requirement per robot. Specifically, we design deterministic algorithms
that attempt to match the time lower bound (Ω(n) rounds) and memory
lower bound (Ω(log n) bits per robot).
Our main result is a deterministic algorithm that is both time and memory optimal, i.e., O(n) rounds and O(log n) bits of memory required
per robot, subject to certain constraints. We subsequently provide results that require less assumptions but are either only time or memory
optimal but not both. We also provide a primitive, utilized often, that
takes robots initially gathered at a node of the ring and disperses them
in a time and memory optimal manner without additional assumptions
required.
