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Black Hole Search: Dynamics, Distribution, and Emergence

Tanvir Kaur, Ashish Saxena, Partha Sarathi Mandal, Kaushik Mondal

TL;DR

An algorithm is presented that solves BHS in the scattered setting using 2\delta_{BH}+17 agents, matching asymptotically the rooted algorithm of \cite{BHS_gen} under the same assumptions, and it does not require knowledge of global parameters or additional model assumptions.

Abstract

A black hole is a malicious node in a graph that destroys any resource entering it without leaving a trace. In the Black Hole Search (BHS) problem with mobile agents, at least one agent must survive and terminate after locating the black hole. Recently, BHS has been studied on 1-bounded 1-interval connected dynamic graphs \cite{BHS_gen}, where a footprint graph exists and at most one edge may disappear per round while connectivity is preserved. Under this model, \cite{BHS_gen} presents an algorithm for the rooted initial configuration, where all agents start from a single node, and proves that at least $2δ_{BH}+1$ agents are necessary in the scattered initial configuration, where agents are arbitrarily placed and $δ_{BH}$ denotes the degree of the black hole. We present an algorithm that solves BHS in the scattered setting using $2δ_{BH}+17$ agents, matching asymptotically the rooted algorithm of \cite{BHS_gen} under the same assumptions. We further investigate the Eventual Black Hole Search (\textsc{Ebhs}) problem, where the black hole may appear at any node and at any time during execution, destroying all agents located there upon its emergence; however, it cannot appear at the home base in round 0, where all agents are initially co-located, and once created, it remains permanently active. While \textsc{Ebhs} has been studied on static rings \cite{Bonnet25}, we extend it to arbitrary static graphs and provide a solution using the minimum number of agents. For rings, our algorithm is optimal in both the number of agents and the running time, and it does not require knowledge of global parameters or additional model assumptions.

Black Hole Search: Dynamics, Distribution, and Emergence

TL;DR

An algorithm is presented that solves BHS in the scattered setting using 2\delta_{BH}+17 agents, matching asymptotically the rooted algorithm of \cite{BHS_gen} under the same assumptions, and it does not require knowledge of global parameters or additional model assumptions.

Abstract

A black hole is a malicious node in a graph that destroys any resource entering it without leaving a trace. In the Black Hole Search (BHS) problem with mobile agents, at least one agent must survive and terminate after locating the black hole. Recently, BHS has been studied on 1-bounded 1-interval connected dynamic graphs \cite{BHS_gen}, where a footprint graph exists and at most one edge may disappear per round while connectivity is preserved. Under this model, \cite{BHS_gen} presents an algorithm for the rooted initial configuration, where all agents start from a single node, and proves that at least agents are necessary in the scattered initial configuration, where agents are arbitrarily placed and denotes the degree of the black hole. We present an algorithm that solves BHS in the scattered setting using agents, matching asymptotically the rooted algorithm of \cite{BHS_gen} under the same assumptions. We further investigate the Eventual Black Hole Search (\textsc{Ebhs}) problem, where the black hole may appear at any node and at any time during execution, destroying all agents located there upon its emergence; however, it cannot appear at the home base in round 0, where all agents are initially co-located, and once created, it remains permanently active. While \textsc{Ebhs} has been studied on static rings \cite{Bonnet25}, we extend it to arbitrary static graphs and provide a solution using the minimum number of agents. For rings, our algorithm is optimal in both the number of agents and the running time, and it does not require knowledge of global parameters or additional model assumptions.
Paper Structure (13 sections, 10 theorems, 2 tables)

This paper contains 13 sections, 10 theorems, 2 tables.

Table of Contents

  1. Introduction
  2. The Model and the Problem
  3. Related Work
  4. Our Contribution
  5. Preliminaries
  6. Algorithm to solve 1-BHS
  7. High-level Idea of our Algorithm
  8. The Algorithm: Algo_Bhs_Scattered
  9. Correctness and Analysis
  10. Emergent Black Hole Search in Static Graphs
  11. The Algorithm: Algo_UXS using Cautious Chain Movement
  12. The correctness and analysis of the algorithm
  13. Conclusion

Key Result

Theorem 1.1

BHS_gen The problem of 1-BHS can be solved by 9 agents starting from a rooted initial configuration in $O(m^2)$ rounds, requiring $O(\log \delta_v)$ storage per node.

Theorems & Definitions (28)

  • definition 1: Kuhn_2010
  • definition 2: GOTOH20211
  • definition 3: Black Hole Dobrev2007BHSRing
  • definition 4: Eventually Emerging Black Hole Bonnet25
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 18 more