Synchronous black hole search in directed graphs

In graph searching, a team of mobile agents aims at clearing the edges of a contaminated graph. To clear an edge, an agent has to slide along it, however, an edge can be recontaminated if there is a path without agents from a contaminated edge to a clear edge. To goal of graph searching is to clear the graph, i.e., all edges are clear simultaneously, using the fewest number of agents. We study this problem in the minimal CORDA model of distributed computation. This model has very weak hypothesis: network nodes and agents are anonymous, have no memory of the past, and agents have no common sense of orientation. Moreover, all agents execute the same algorithm in the Look-Compute-Move manner and in an asynchronous environment. One interest of this model is that, if the clearing can be done by the agents starting from arbitrary positions (e.g., after faults or recontamination), the lack of memory implies that the clearing is done perpetually and then provides a first approach of fault-tolerant graph searching. Constraints due to the minimal CORDA model lead us to define a new variant of graph searching, called graph searching without collisions, where more than one agent cannot occupy the same node. We show that, in a centralized setting, this variant does not have the same behavior as classical graph searching. For instance, it not monotonous nor close by subgraph. We show that, in a graph with maximum degree ∆, the smallest number of agents required to clear a graph without collisions is at most ∆ times the number of searchers required when collisions are allowed. Moreover, this bound is tight up to a constant ratio. Then, we fully characterize graph searching without collisions in trees. In a distributed setting, i.e., in the minimal CORDA model, the question we ask is the following. Given a graph G, does there exist an algorithm that clears G, whatever be the initial positions of the agents on distinct vertices. In the case of a path network, we show that it is not possible is the number of agents is even in a path of odd order, or if there are at most two agents in a path with at least three vertices. We present an algorithm that clears all paths in all remaining cases. Finally, we propose an algorithm that clears any tree using a sufficient number of agents.

Combinatorics, Probability and Computing, 2007

A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node, without leaving any trace. We consider the task of locating a black hole in a (partially) synchronous tree network, assuming an upper bound on the time of any edge traversal by an agent. The minimum number of agents capable of identifying a black hole is two. For a given tree and given starting node we are interested in the fastest possible black hole search by two agents. For arbitrary trees we give a 5/3approximation algorithm for this problem. We give optimal black hole search algorithms for two "extreme" classes of trees: the class of lines and the class of trees in which any internal node (including the root which is the starting node) has at least 2 children.

Lecture Notes in Computer Science, 2012

We propose optimal (w.r.t. the number of robots) solutions for the deterministic terminating exploration (exploration for short) of a grid-shaped network by a team of k asynchronous oblivious robots in the asynchronous non-atomic model, so-called CORDA.

Journal of Graph Theory, 2015

Let G be a graph of minimum degree at least two with no induced subgraph isomorphic to K 1,6 . We prove that if G is not isomorphic to one of eight exceptional graphs, then it is possible to assign twoelement subsets of {1, 2, 3, 4, 5} to the vertices of G in such a way that for every i ∈ {1, 2, 3, 4, 5} and every vertex v ∈ V (G) the label i is assigned to v or one of its neighbors. It follows that G has fractional domatic number at least 5/2. This is motivated by a problem in robotics and generalizes a result of Fujita, Yamashita and Kameda who proved that the same conclusion holds for all 3-regular graphs.

Lecture Notes in Computer Science, 2015

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots perpetually move along the domain, represented by all points belonging to the graph edges, not exceeding their maximal speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e. they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolmen? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by f k (G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing f k (G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study the case of general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness f k (G) = (f + 1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs-a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs.

arXiv (Cornell University), 2020

This work focuses on the following question related to the Gathering problem of n autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of robots that do not agree on any axis of their coordinate systems (disoriented robots) and see other robots only up to a constant distance (limited visibility) in o(n 2) fully synchronous rounds (the Fsync scheduler)? The best known algorithm that solves Gathering of disoriented robots with limited visibility in the OBLOT model (oblivious robots) needs Θ n 2 rounds [8]. The lower bound for this algorithm even holds in a simplified closed chain model, where each robot has exactly two neighbors and the chain connections form a cycle. The only existing algorithms achieving a linear number of rounds for disoriented robots assume robots that are located on a two dimensional grid [1] and [7]. Both algorithms make use of locally visible lights to communicate state information (the LUMIN OU S model). In this work, we show for the closed chain model, that n disoriented robots with limited visibility in the Euclidean plane can be gathered in Θ (n) rounds assuming the LUMIN OU S model. The lights are used to initiate and perform so-called runs along the chain. For the start of such runs, locally unique robots need to be determined. In contrast to the grid [1], this is not possible in every configuration in the Euclidean plane. Based on the theory of isogonal polygons by Branko Grünbaum, we identify the class of isogonal configurations in which-due to a high symmetry-no such locally unique robots can be identified. Our solution combines two algorithms: The first one gathers isogonal configurations; it works without any lights. The second one works for non-isogonal configurations; it identifies locally unique robots to start runs, using a constant number of lights. Interleaving these algorithms solves the Gathering problem in O (n) rounds. * This paper is a full version of the brief announcement presented at SSS 2020.

Information Processing Letters, 2011

We consider the problem of exploring an anonymous line by a team of k identical, oblivious, asynchronous deterministic mobile robots that can view the environment but cannot communicate. We completely characterize sizes of teams of robots capable of exploring an n-node line. For kn, exploration by k robots turns out to be possible, if and only if either k=3, or

arXiv (Cornell University), 2020

Imagine that a swarm of robots is given, these robots must communicate with each other, and they can do so if certain conditions are met. We say that the swarm is connected if there is at least one way to send a message between each pair of robots. A robot can move from a work station to another only if the connectivity of the swarm is preserved in order to perform some tasks. We model the problem via graph theory, we study connected subgraphs and how to motion them inside a connected graph preserving the connectivity. We determine completely the group of movements.

2008

A team of identical, oblivious, mobile agents (robots) has to explore an anonymous unoriented tree by visiting all its nodes. Robots start from arbitrary different nodes of the tree and operate in Look-Compute-Move cycles. At the end, every node must be visited by at least one robot, and all robots must stop. In one cycle, a robot takes a snapshot of the current configuration (Look), makes a decision to stay idle or to move to one of its adjacent nodes (Compute), and in the latter case makes an instantaneous move to this neighbor (Move). Cycles are performed asynchronously for each robot. We present an exploration algorithm for arbitrary n-node trees of maximum degree 3 using O(log n/ log log n) robots, and we prove that for some such trees Ω(log n/ log log n) robots are necessary to explore them. We also show that in order to explore some n-node trees of maximum degree 4, Ω(n) robots are necessary. By contrast we show that in order to explore trees that do not have non-trivial auto...

Algorithms for Sensor Systems, 2019

We consider (n, f)-search on a circle, a search problem of a hidden exit on a circle of unit radius for n > 1 robots, f of which are faulty. All the robots start at the centre of the circle and can move anywhere with maximum speed 1. During the search, robots may communicate wirelessly. All messages transmitted by all robots are tagged with the robots' unique identifiers which cannot be corrupted. The search is considered complete when the exit is found by a non-faulty robot (which must visit its location) and the remaining non-faulty robots know the correct location of the exit. We study two models of faulty robots. First, crash-faulty robots may stop operating as instructed, and thereafter they remain nonfunctional. Second, Byzantine-faulty robots may transmit untrue messages at any time during the search so as to mislead the non-faulty robots, e.g., lie about the location of the exit. When there are only crash fault robots, we provide optimal algorithms for the (n, f)-search problem, with optimal worst-case search completion time 1 + (f +1)2π n. Our main technical contribution pertains to optimal algorithms for (n, 1)-search with a Byzantine-faulty robot, minimizing the worst-case search completion time, which equals 1 + 4π n .