Black Hole Search by Scattered Agents in Dynamic Rings

TL;DR

This paper investigates the problem of locating a black hole in a dynamic ring network using scattered agents and movable pebbles, establishing optimal solutions with minimal agents and moves.

Contribution

It introduces an optimal algorithm using three agents and movable pebbles for black hole search in dynamic rings, proving both sufficiency and necessity of the resources.

Findings

Three agents suffice to find the black hole in O(n^2) moves.

Three agents are proven to be optimal for this problem.

The O(n^2) move complexity is tight, even with enhanced communication.

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 $\Omega(n^2)$ moves for any algorithm solving the problem with three agents, even when stronger communication mechanisms, such as unlimited-size whiteboards on nodes, are available.