Efficient Dispersion on an Anonymous Ring in the Presence of Weak Byzantine Robots
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 selfdriving 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 wellstudied when robots are nonfaulty. 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 ($\Omega(n)$ rounds) and memory lower bound ($\Omega(\log n)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.11439
 Bibcode:
 2020arXiv200411439R
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Data Structures and Algorithms
 EPrint:
 20 pages