A Black Hole is an harmful host in a network that destroys incoming agents without leaving any trace of such event. The problem of locating the black hole in a network through a team of agent coordinated by a common protocol is usually referred in literature as the Black Hole Search problem (or BHS for brevity) and it is a consolidated research topic in the area of distributed algorithms. The aim of this paper is to extend the results for BHS by considering more general (and hence harder) classes of dangerous host. In particular we introduce rB-hole as a probabilistic generalization of the Black Hole, in which the destruction of an incoming agent is a purely random event happening with some fixed probability (like flipping a biased coin). The main result we present is that if we tolerate an arbitrarily small error probability in the result then the rB-hole Search problem, or RBS, is not harder than the usual BHS. We establish this result in two different communication model, specifically both in presence or absence of whiteboards non-located at the homebase. The core of our methods is a general reduction tool for transforming algorithms for the black hole into algorithms for the rB-hole.
Deep Dive into Searching for a dangerous host: randomized vs. deterministic.
A Black Hole is an harmful host in a network that destroys incoming agents without leaving any trace of such event. The problem of locating the black hole in a network through a team of agent coordinated by a common protocol is usually referred in literature as the Black Hole Search problem (or BHS for brevity) and it is a consolidated research topic in the area of distributed algorithms. The aim of this paper is to extend the results for BHS by considering more general (and hence harder) classes of dangerous host. In particular we introduce rB-hole as a probabilistic generalization of the Black Hole, in which the destruction of an incoming agent is a purely random event happening with some fixed probability (like flipping a biased coin). The main result we present is that if we tolerate an arbitrarily small error probability in the result then the rB-hole Search problem, or RBS, is not harder than the usual BHS. We establish this result in two different communication model, specifical
The Black Hole Search problem, or BHS, [3,7,5,4,8,2,6] has recently gained a lot of interest among the research community in mobile and distributed computation. A Black Hole represents a "malicious" host in a network, which destroys every agent that tries to pass through it. No trace of such destruction event will be observable by any other agent. The BHS problem requires to find a strategy to coordinate a set of autonomous and mobile agents in order to discover and correctly report the location of the Black Hole inside a network. A correct solution is required to terminate after a finite amount of moves with at least one of the agents surviving and reporting the correct output.
Several authors have investigated the BHS problem under different hypothesis about network’s topology (like ring [4], mesh, hypercube, etc. [3,6,2]), kind of communication devices (i.e., tokens instead of whiteboard [7]), network’s topological knowledge [6], presence of multiple black holes [1], etc. These different algorithms (or protocols) are usually compared on the basis of two main complexity measures: the number of moves performed and the number of agents required, where both this parameters are taken in the worst case.
In this paper we address the malicious host question in a more general form, namely we introduce the concept of rB-Hole, which is a randomized generalization of the Black Hole, and then study new strategies for its localization in a network. We will see that the rB-Hole Search problem ( rBHS for brevity) problem can be resolved only if we tolerate an error probability in the output. Under this hypothesis the rBHS problem is solvable and we will provide a general technique to derive an algorithm for rBHS from an algorithm for BHS. As a main consequence, by applying our technique to some of the standard result about the BHS case [4,3], we provide generalizations of these methods to work for the rBHS problem without increasing asymptotically the number of moves performed or the number of agents required.
This section is dedicated to introducing the model of computation and useful background about the problem. The term agent denotes a computational entity allowed to perform an arbitrary computation. The agents are equipped with a local bounded memory which maintains the status of their computations or other useful information. They are able to move themselves in the network by following the links connecting adjacent nodes. Moreover, the agents can communicate by reading from and writing on shared memory units located on the nodes, called whiteboards. Access to a whiteboard is done in mutual exclusion. We assume that the amount of storage available on a whiteboard is O(log n) bits.
It is important to notice that the agents are asynchronous, this means there is no assumption on the time taken by an agent to perform a generic action, like a move on a link or a computation step. This implies the impossibility to predict when one of this action will eventually end.
In the following the network will be represented by a connected undirected graph G, whose nodes can be anonymous (i.e., without unique names).
As defined in [4], a black hole is a stationary process located at a node, which destroys any agent arriving at that node. No observable trace of such destruction event will be evident to other agents. The Black Hole Search (BHS) problem [4] consists of devising a strategy for coordinating the agents in order to discover the position of the black hole in a network.
At the beginning of the strategy all the agents are assumed to be co-located in a unique safe node called homebase. After a finite number of moves, at least one of the agent must survive and be able to indicate the position of the black hole in the network.
In this paper we propose a generalization of the BHS problem by introducing the notion of rBhole. A rB-hole is an aleatory process located on an host which can destroy visiting agents with some fixed probability p. More precisely the interaction between a visiting agent and the rB-hole can be schematized as follows:
The agent move on a link from a safe node to the rB-hole. The rB-hole flip a biased coin which give HEAD with probability p, where p is a parameter of the rB-hole. If an HEAD comes out the agent is killed otherwise he advances to the next phase.
The agent enters the rB-hole and gains access to its internal whiteboard. The agent is now safe and hence he is able to consistently modify the whiteboard.
The agent moves on a link from the rB-hole to a safe node. This phase is symmetrical to the first; a biased coin is flipped and the agent is killed with probability p, otherwise he safely leaves the rB-hole.
Observe that a black hole is simply a rB-hole with p equal to 1. The rBHS problem is defined as the analogous of the BHS problem for the rB-hole.
Since the rBHS is indeed a generalization of the BHS problem, it automatically inherits all of its known lower bounds. The follo
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