Black Hole Search by Scattered Agents in Dynamic Rings
Giuseppe Antonio Di Luna, Paola Flocchini, Giuseppe Prencipe, Nicola Santoro
TL;DR
This work addresses locating a silent black hole in a dynamic 1-interval connected ring with scattered agents. It proves that three agents suffice to locate the Bh in $O(n^{2})$ moves and that this bound is tight under exogenous communication, establishing a $\Omega(n^{2})$ lower bound for any size-optimal algorithm and providing a matching Gather&Locate strategy. It also shows that endogenous communication (Face-to-Face) cannot solve the problem for scattered agents, highlighting the crucial role of external aids like pebbles or whiteboards. Together, these results delineate the cost of dispersion and the limits of different communication models in black hole search on dynamic networks, and they extend the understanding of Bh search beyond colocated scenarios into scattered-agent settings.
Abstract
In this paper, we address the challenge of locating a black hole within a dynamic graph using a set of scattered agents, which start from arbitrary positions in the graph. A black hole is defined as a node that silently eliminates any agent that visits it, effectively modeling network failures such as a crashed host or a destructive virus. The black hole search problem is considered solved when at least one agent survives and possesses a complete map of the graph, including the precise location of the black hole. Our study focuses on the scenario where the underlying graph is a dynamic 1-interval connected ring: a ring graph where, in each round, one edge may be absent. Agents communicate with other agents using movable pebbles that can be placed on nodes. In this setting, we demonstrate that three agents are sufficient to identify the black hole in $O(n^2)$ moves. Furthermore, we prove that this number of agents is optimal. Additionally, we establish that the complexity bound is tight, requiring $Ω(n^2)$ moves for any algorithm solving the problem with three agents, even when stronger communication mechanisms, such as unlimited-size whiteboards on nodes, are available.