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 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 ((n) rounds) and memory lower bound ((logn)
bits per robot).
Our main result is a deterministic algorithm that is both time and memory optimal, i.e.,
O(n) rounds and O(logn) 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.
