Black Hole Search by Scattered Agents in Dynamic Rings

2024-04-23

AI Summary

This paper investigates the black hole search problem—locating a node that destroys any agent visiting it—in dynamic 1-interval-connected ring graphs, where at most one edge may be missing per round. Agents start at arbitrary nodes and communicate via mobile tokens (pebbles) with limited capabilities. We establish, for the first time, that three agents are both necessary and sufficient to deterministically locate the black hole in this model. We present a deterministic distributed algorithm achieving O(n²) move complexity and prove Ω(n²) as a tight lower bound—even when augmenting communication (e.g., with whiteboards)—demonstrating that no algorithm can improve upon quadratic time. Our results resolve two fundamental open problems in dynamic ring networks: determining the minimum number of agents required for black hole search and establishing the optimal time complexity.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Deterministic Planning
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Dynamic Graphs
• Black Hole Detection
• Minimization of Helpers

Innovation

• Dynamic 1-Interval Connected Ring
• Minimum Three Helpers
• O(n^2) Steps