Black Hole Search: Dynamics, Distribution, and Emergence

TL;DR

This paper advances black hole search algorithms in dynamic graphs, providing solutions for scattered initial configurations and the Eventual Black Hole Search problem in static graphs, with improved agent efficiency and minimal assumptions.

Contribution

It introduces an algorithm for BHS with 2δ_{BH}+17 agents in scattered initial setups and extends Eventual Black Hole Search to arbitrary static graphs with four agents, requiring no global knowledge.

Findings

Matching asymptotic efficiency with rooted algorithms in dynamic graphs

Solution for Eventual Black Hole Search in arbitrary static graphs with four agents

No need for global parameters in the Eventual Black Hole Search algorithm

Abstract

A black hole is a malicious node in a graph that destroys resources entering into it without leaving any trace. The problem of Black Hole Search (BHS) using mobile agents requires that at least one agent survives and terminates after locating the black hole. Recently, this problem has been studied on 1-bounded 1-interval connected dynamic graphs \cite{BHS_gen}, where there is a footprint graph, and at most one edge can disappear from the footprint in a round, provided that the graph remains connected. In this setting, the authors in \cite{BHS_gen} proposed an algorithm that solves the BHS problem when all agents start from a single node (rooted initial configuration). They also proved that at least $2\delta_{BH} + 1$ agents are necessary to solve the problem when agents are initially placed arbitrarily across the nodes of the graph (scattered initial configuration), where $\delta_{BH}$ denotes the degree of the black hole. In this work, we present an algorithm that solves the BHS problem using $2\delta_{BH} + 17$ initially scattered agents. Our result matches asymptotically with the rooted algorithm of \cite{BHS_gen} under the same model assumptions. Further, we study the Eventual Black Hole Search (\textsc{Ebhs}) problem, in which the black hole may appear at any node and at any time during the execution of the algorithm, destroying all agents located on that node at the time of its appearance. However, the black hole cannot emerge at the home base in round~0, where the home base is the node at which all agents are initially co-located. Once the black hole appears, it remains active at that node for the rest of the execution. This problem has been studied on static rings~\cite{Bonnet25}; here we extend it to arbitrary static graphs and provide a solution using four agents. Moreover, it does not require any knowledge of global parameters or additional model assumptions.