Black Hole Search in Dynamic Graphs

TL;DR

This work extends Black Hole Search to arbitrary dynamic graphs where an adversary may delete edges each round while preserving connectivity. It develops both impossibility results and constructive algorithms under rooted and arbitrary initial configurations, with memory-limited agents and log-sized whiteboards. A key contribution is the $O(|E|^2)$-time, nine-agent rooted 1-BHS algorithm, improved to six agents, along with an impossibility for $f$-BHS with $2f+1$ agents and a $6f$-agent rooted algorithm solving $f$-BHS in exponential time. Overall, the results advance understanding of exploration and BH identification in dynamic networks, offering practical protocols under weak sensing and communication assumptions.

Abstract

A black hole is considered to be a dangerous node present in a graph that disposes of any resources that enter that node. Therefore, it is essential to find such a node in the graph. Let a group of agents be present on a graph $G$. The Black Hole Search (BHS) problem aims for at least one agent to survive and terminate after {finding} the black hole. This problem is already studied for specific dynamic graph classes such as rings, cactuses, and tori {where finding the black hole means at least one agent needs to survive and terminate after knowing at least one edge associated with the black hole. In this work, we investigate the problem of BHS for general graphs.} In the dynamic graph, adversary may remove edges at each round keeping the graph connected. We consider two cases: (a) at any round at most one edge can be removed (b) at any round at most $f$ edges can be removed. For both scenarios, we study the problem when the agents start from a rooted initial configuration. We consider each agent has $O(\log n)$ memory and each node has $O(\log n)$ storage. For case (a), we present an algorithm with $9$ agents that solves the problem of BHS in $O(|E|^2)$ time where $|E|$ is the number of edges and $δ_v$ is the degree of the node $v$ in $G$. We show it is impossible to solve for $2δ_{BH}$ many agents starting from an arbitrary configuration where $δ_{BH}$ is the degree of the black hole in $G$. We also provide another improved algorithm that uses $6$ agents from a rooted initial configuration to solve the problem of BHS. For case (b), we provide an algorithm using $6f$ agents to solve the problem of BHS, albeit taking exponential time. We also provide an impossibility result for $2f+1$ agents starting from a rooted initial configuration. This result holds even if unlimited storage is available on each node and the agents have infinite memory.

Figures (8)

• Figure 1: Construction of the class of graph for impossibility of $1-$BHS
• Figure 2: Flow Chart representing cautious walk by a leader ($a_1$) and helpers ($a_1'$ and $a_1"$)
• Figure 3: Construction of the graph $K_5$ when $f=3$ for impossibility of $f-$BHS
• Figure 4: (a) Let TVG $\mathcal{G}$ has no black hole and agents $A_1$ and $A_2$ be initially positioned at a node $v_r$ (root node). Let the agents reach $u$, and they encounter a missing edge, (b) $A_1$ waits at $u$ but $A_2$ skips the edge $(u,v)$ and reaches at $v_r$ within $4|E|$ rounds if the edge $(u,v)$ is missing continuously for $4|E|$ rounds. Within another (at most) $4|E|$ rounds $A_2$ explores $G\setminus(u,v)$ if edge $(u,v)$ is missing, (c) If the edge $(u,v)$ appears before $A_2$ could explore the graph $G\setminus(u,v)$, $A_1$ moves a step forward.
• Figure 5: (a) Let $A_1$ and $A_2$ be present at $u_1$ in the initial configuration. (b) Both agents move as $u_1 \rightarrow u_2\rightarrow u_3$ and then encounter a missing edge. (c) $A_1$ waits at $u_3$ for the missing edge to reappear and $A_2$ moves as $u_3 \rightarrow u_2\rightarrow u_1 \rightarrow u_4$. It attempts to move via edge $(u_4,u_3)$ but finds it missing, so $A_2$ skips it. Thus, it moves to reach $u_5$ and the exploration of the graph is done when the edge $(u_3,u_4)$ remains missing. (d) If $A_2$ finds a missing edge at $(u_1,u_4)$, then $A_1$ proceeds forward and reaches $u_4$. Since the movement by $A_2$ is continued even when it encounters a missing edge, this guarantees exploration of the graph.

Theorems & Definitions (27)

• definition 1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• remark 1