Black Hole Search with Finite Automata Scattered in a Synchronous Torus

TL;DR

This paper addresses the black hole search problem in oriented torus networks using finite automata with constant tokens, establishing lower bounds and providing an optimal deterministic solution.

Contribution

It introduces the first finite automata-based solution for black hole search in torus networks, with resource bounds independent of network size.

Findings

Finite automata cannot solve the problem without movable tokens.

Lower bounds on agents and tokens needed for solution.

Deterministic solution using minimal resources.

Abstract

We consider the problem of locating a black hole in synchronous anonymous networks using finite state agents. A black hole is a harmful node in the network that destroys any agent visiting that node without leaving any trace. The objective is to locate the black hole without destroying too many agents. This is difficult to achieve when the agents are initially scattered in the network and are unaware of the location of each other. Previous studies for black hole search used more powerful models where the agents had non-constant memory, were labelled with distinct identifiers and could either write messages on the nodes of the network or mark the edges of the network. In contrast, we solve the problem using a small team of finite-state agents each carrying a constant number of identical tokens that could be placed on the nodes of the network. Thus, all resources used in our algorithms are independent of the network size. We restrict our attention to oriented torus networks and first show that no finite team of finite state agents can solve the problem in such networks, when the tokens are not movable. In case the agents are equipped with movable tokens, we determine lower bounds on the number of agents and tokens required for solving the problem in torus networks of arbitrary size. Further, we present a deterministic solution to the black hole search problem for oriented torus networks, using the minimum number of agents and tokens.