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Black Hole Search in Dynamic Tori

Adri Bhattacharya, Giuseppe F. Italiano, Partha Sarathi Mandal

TL;DR

This work addresses the black hole search problem on a dynamic torus with 1-interval connectivity, integrating two initial agent configurations (co-located and scattered). It derives lower bounds on the required number of agents and the termination time, and then presents two scalable algorithms for each configuration that achieve near-optimal round complexity: $O(nm^{1.5})$ with $k=n+3$ (co-located) and $O(nm)$ with $k=n+4$, plus $O(nm^{1.5})$ with $k=n+6$ and $O(nm)$ with $k=n+7$ (scattered). Central to the methods are cautious-walk strategies and whiteboard-based communication that enable safe exploration of rings and systematic black hole localization, even under adversarial edge dynamics. The results advance BHS in dynamic networks by providing tight agent-count vs. round trade-offs and by illustrating near-optimal, scalable strategies for both co-located and scattered initial configurations.

Abstract

We investigate the black hole search problem by a set of mobile agents in a dynamic torus. Black hole is defined to be a dangerous stationary node which has the capability to destroy any number of incoming agents without leaving any trace of its existence. A torus of size $n\times m$ ($3\leq n \leq m$) is a collection of $n$ row rings and $m$ column rings, and the dynamicity is such that each ring is considered to be 1-interval connected, i.e., in other words at most one edge can be missing from each ring at any round. The parameters which define the efficiency of any black hole search algorithm are: the number of agents and the number of rounds (or \textit{time}) for termination. We consider two initial configurations of mobile agents: first, the agents are co-located and second, the agents are scattered. In each case, we establish lower and upper bounds on the number of agents and on the amount of time required to solve the black hole search problem.

Black Hole Search in Dynamic Tori

TL;DR

This work addresses the black hole search problem on a dynamic torus with 1-interval connectivity, integrating two initial agent configurations (co-located and scattered). It derives lower bounds on the required number of agents and the termination time, and then presents two scalable algorithms for each configuration that achieve near-optimal round complexity: with (co-located) and with , plus with and with (scattered). Central to the methods are cautious-walk strategies and whiteboard-based communication that enable safe exploration of rings and systematic black hole localization, even under adversarial edge dynamics. The results advance BHS in dynamic networks by providing tight agent-count vs. round trade-offs and by illustrating near-optimal, scalable strategies for both co-located and scattered initial configurations.

Abstract

We investigate the black hole search problem by a set of mobile agents in a dynamic torus. Black hole is defined to be a dangerous stationary node which has the capability to destroy any number of incoming agents without leaving any trace of its existence. A torus of size () is a collection of row rings and column rings, and the dynamicity is such that each ring is considered to be 1-interval connected, i.e., in other words at most one edge can be missing from each ring at any round. The parameters which define the efficiency of any black hole search algorithm are: the number of agents and the number of rounds (or \textit{time}) for termination. We consider two initial configurations of mobile agents: first, the agents are co-located and second, the agents are scattered. In each case, we establish lower and upper bounds on the number of agents and on the amount of time required to solve the black hole search problem.
Paper Structure (17 sections, 10 theorems, 1 equation, 5 figures, 2 tables)

This paper contains 17 sections, 10 theorems, 1 equation, 5 figures, 2 tables.

Table of Contents

  1. Our Contribution
  2. Model and Problem Definition
  3. Graph Model
  4. Agent Model
  5. Configuration
  6. Lower Bound Results
  7. Co-located Agents
  8. Scattered Agents
  9. Preliminaries
  10. Subroutines
  11. Co-located Agents
  12. BHS with $n+3$ agents
  13. BHS with $n+4$ agents
  14. Scattered Agents
  15. BHS with $n+6$ agents
  16. ...and 2 more sections

Key Result

Theorem 1

Given a dynamic torus $\mathcal{G}$ of size $n \times m$, there does not exist a BHS algorithm which correctly locates the black hole with $k=n+1$ co-located agents and each node in $\mathcal{G}$ contains a whiteboard of $O(1)$ bits.

Figures (5)

  • Figure 1: A $3\times4$ dynamic torus, where dashed edges represents a disappeared or missing edge
  • Figure 2: A initial configuration when the agents are scattered along the dynamic torus
  • Figure 3: An instance where 4 agents are operating along a ring while executing Algorithm \ref{['alg-1']}
  • Figure 4: Representing all possible black hole consumption while executing either algorithm \ref{['alg-4']} or \ref{['alg-5']}
  • Figure 5: Represents all possible black hole consumption while executing either Algorithm \ref{['alg-6']} or \ref{['alg-7']}

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2: BHSDynRingLuna
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10