Perpetual Exploration of a Ring in Presence of Byzantine Black Hole

TL;DR

The main results focus on minimizing the agent number while ensuring that perpetual exploration is achieved even in presence of such a node under various communication models and starting positions.

Abstract

Perpetual exploration is a fundamental problem in the domain of mobile agents, where an agent needs to visit each node infinitely often. This issue has received lot of attention, mainly for ring topologies, presence of black holes adds more complexity. A black hole can destroy any incoming agent without any observable trace. In \cite{BampasImprovedPeriodicDataRetrieval,KralovivcPeriodicDataRetrievalFirst}, the authors considered this problem in the context of \textit{ Periodic data retrieval}. They introduced a variant of black hole called gray hole (where the adversary chooses whether to destroy an agent or let it pass) among others and showed that 4 asynchronous and co-located agents are essential to solve this problem (hence perpetual exploration) in presence of such a gray hole if each node of the ring has a whiteboard. This paper investigates the exploration of a ring in presence of a ``byzantine black hole''. In addition to the capabilities of a gray hole, in this variant, the adversary chooses whether to erase any previously stored information on that node. Previously, one particular initial scenario (i.e., agents are co-located) and one particular communication model (i.e., whiteboard) are investigated. Now, there can be other initial scenarios where all agents may not be co-located. Also, there are many weaker models of communications (i.e., Face-to-Face, Pebble) where this problem is yet to be investigated. The agents are synchronous. The main results focus on minimizing the agent number while ensuring that perpetual exploration is achieved even in presence of such a node under various communication models and starting positions. Further, we achieved a better upper and lower bound result (i.e., 3 agents) for this problem (where the malicious node is a generalized version of a gray hole), by trading-off scheduler capability, for co-located and in presence of a whiteboard.

Figures (2)

• Figure 1: An execution of PerpExplore-Coloc-Pbl, starting from the configuration where $a_0$ and $a_1$ are together on a vertex, to the configuration where $a_0$ and $a_1$ are on the same vertex again, which is the clockwise neighbour of the earlier vertex. The red node is the byzantine black hole, the green node is the $home$ and red boxes are pebbles.
• Figure 2: (a)$h_i$'s are $home$ marked as green. Agent $a_i$ is on $h_i$ initially with a pebble. We name the pebble initially at $h_i$ as $p_i$, but in reality they are anonymous. (a-b) Each agent moves clockwise until the next $home$ (i.e., $h_{(i+1) \pmod{4}}$) without carrying any pebble. The agents already reached ( here $a_1$ and $a_3$) waits for others. (c-e) All agents are on their clockwise nearest $home$. They start moving counter clockwise together with the pebble present at their current location towards their initial $home$. The agents which already reaches their $home$, wait for others to reach their $home$.

Theorems & Definitions (32)

• Definition 1: PerpExploration-BBH
• Theorem 2
• Corollary 3
• Corollary 4
• Theorem 5
• Corollary 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10