Rendezvous of Distance-aware Mobile Agents in Unknown Graphs

We study the problem of rendezvous of two mobile agents starting at distinct locations in an unknown graph. The agents have distinct labels and walk in synchronous steps. However the graph is unlabelled and the agents have no means of marking the nodes of the graph and cannot communicate with or see each other until they meet at a node. When the graph is very large we want the time to rendezvous to be independent of the graph size and to depend only on the initial distance between the agents and some local parameters such as the degree of the vertices, and the size of the agent's label. It is well known that even for simple graphs of degree $\Delta$, the rendezvous time can be exponential in $\Delta$ in the worst case. In this paper, we introduce a new version of the rendezvous problem where the agents are equipped with a device that measures its distance to the other agent after every step. We show that these \emph{distance-aware} agents are able to rendezvous in any unknown graph, in time polynomial in all the local parameters such the degree of the nodes, the initial distance $D$ and the size of the smaller of the two agent labels $l = \min(l_1, l_2)$. Our algorithm has a time complexity of $O(\Delta(D+\log{l}))$ and we show an almost matching lower bound of $\Omega(\Delta(D+\log{l}/\log{\Delta}))$ on the time complexity of any rendezvous algorithm in our scenario. Further, this lower bound extends existing lower bounds for the general rendezvous problem without distance awareness

Byzantine Gathering in Networks

This paper investigates an open problem introduced in [14]. Two or more mobile agents start from different nodes of a network and have to accomplish the task of gathering which consists in getting all together at the same node at the same time. An adversary chooses the initial nodes of the agents and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label but does not know the labels of the other agents or their positions relative to its own. Agents move in synchronous rounds and can communicate with each other only when located at the same node. Up to f of the agents are Byzantine. A Byzantine agent can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. What is the minimum number M of good agents that guarantees deterministic gathering of all of them, with termination? We provide exact answers to this open problem by considering the case when the agents initially know the size of the network and the case when they do not. In the former case, we prove M=f+1 while in the latter, we prove M=f+2. More precisely, for networks of known size, we design a deterministic algorithm gathering all good agents in any network provided that the number of good agents is at least f+1. For networks of unknown size, we also design a deterministic algorithm ensuring the gathering of all good agents in any network but provided that the number of good agents is at least f+2. Both of our algorithms are optimal in terms of required number of good agents, as each of them perfectly matches the respective lower bound on M shown in [14], which is of f+1 when the size of the network is known and of f+2 when it is unknown

Rendezvous on a Line by Location-Aware Robots Despite the Presence of Byzantine Faults

A set of mobile robots is placed at points of an infinite line. The robots are equipped with GPS devices and they may communicate their positions on the line to a central authority. The collection contains an unknown subset of "spies", i.e., byzantine robots, which are indistinguishable from the non-faulty ones. The set of the non-faulty robots need to rendezvous in the shortest possible time in order to perform some task, while the byzantine robots may try to delay their rendezvous for as long as possible. The problem facing a central authority is to determine trajectories for all robots so as to minimize the time until the non-faulty robots have rendezvoused. The trajectories must be determined without knowledge of which robots are faulty. Our goal is to minimize the competitive ratio between the time required to achieve the first rendezvous of the non-faulty robots and the time required for such a rendezvous to occur under the assumption that the faulty robots are known at the start. We provide a bounded competitive ratio algorithm, where the central authority is informed only of the set of initial robot positions, without knowing which ones or how many of them are faulty. When an upper bound on the number of byzantine robots is known to the central authority, we provide algorithms with better competitive ratios. In some instances we are able to show these algorithms are optimal

Asynchronous rendezvous with different maps

© Springer Nature Switzerland AG 2019. This paper provides a study on the rendezvous problem in which two anonymous mobile entities referred to as robots rA and rB are asked to meet at an arbitrary node of a graph G = (V,E). As opposed to more standard assumptions robots may not be able to visit the entire graph G. Namely, each robot has its own map which is a connected subgraph of G. Such mobility restrictions may be dictated by the topological properties combined with the intrinsic characteristics of robots preventing them from visiting certain edges in E. We consider four different variants of the rendezvous problem introduced in [Farrugia et al. SOFSEM’15] which reflect on restricted maneuverability and navigation ability of rA and rB in G. In the latter, the focus is on models in which robots’ actions are synchronised. The authors prove that one of the maps must be a subgraph of the other. I.e., without this assumption (or some extra knowledge) the rendezvous problem does not have a feasible solution. In this paper, while we keep the containment assumption, we focus on asynchronous robots and the relevant bounds in the four considered variants. We provide some impossibility results and almost tight lower and upper bounds when the solutions are possible

Gathering of Robots on Anonymous Grids without multiplicity detection

International audienceThe paper studies the gathering problem on grid networks. A team of robots placed at different nodes of a grid, have to meet at some node and remain there. Robots operate in Look-Compute-Move cycles; in one cycle, a robot perceives the current configuration in terms of occupied nodes (Look), decides whether to move towards one of its neighbors (Compute), and in the positive case makes the computed move instantaneously (Move). Cycles are performed asynchronously for each robot. The problem has been deeply studied for the case of ring networks. However, the known techniques used on rings cannot be directly extended to grids. Moreover, on rings, another assumption concerning the so-called multiplicity detection capability was required in order to accomplish the gathering task. That is, a robot is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. In this paper, we provide a full characterization about gatherable configurations for grids. In particular, we show that in this case, the multiplicity detection is not required. Very interestingly, sometimes the problem appears trivial, as it is for the case of grids with both odd sides, while sometimes the involved techniques require new insights with respect to the well-studied ring case. Moreover, our results reveal the importance of a structure like the grid that allows to overcome the multiplicity detection with respect to the ring case

How to gather asynchronous oblivious robots on anonymous rings

A set of robots arbitrarily placed on the nodes of an anonymous graph have to meet at one common node and remain in there. This problem is known in the literature as the \emph{gathering}. Robots operate in Look-Compute-Move cycles; in one cycle, a robot takes a snapshot of the current configuration (Look), decides whether to stay idle or to move to one of its neighbors (Compute), and in the latter case makes the computed move instantaneously (Move). Cycles are performed asynchronously for each robot. Moreover, each robot is empowered by the so called \emph{multiplicity detection} capability, that is, a robot is able to detect during its Look operation whether a node is empty, or occupied by one robot, or occupied by an undefined number of robots greater than one. The described problem has been extensively studied during the last years. However, the known solutions work only for specific initial configurations and leave some open cases. In this paper, we provide an algorithm which solves the general problem, and is able to detect all the non-gatherable configurations. It is worth noting that our new algorithm makes use of a unified and general strategy for any initial configuration.Un ensemble de robots placés arbitrairement sur les sommets d'un graphe anonyme doivent se rencontrer sur un sommet commun. Ce problème est connu dans la littérature comme le \emph{gathering}. Les robots utilisent des cycles Look-Compute-Move; dans un cycle, un robot prend un instantané de la configuration actuelle (Look), décide de rester inactif ou de se déplacer sur l'un de ses voisins (Compute), et dans ce cas, fait le déplacement (Move). Les cycles sont exécutés de manière asynchrone pour chaque robot. Chaque robot possède la capacité de \emph{multiplicity detection}: un robot est capable de détecter au cours de son opération Look si un sommet est vide, occupé par un robot, ou occupé par un nombre indéfini de robots. Le problème décrit a été largement étudié au cours des dernières années. Toutefois, les solutions connues ne sont valides que pour des configurations initiales spécifiques. Nous fournissons un algorithme qui résout le problème général, et est capable de détecter toutes les configurations initiales pour lesquelles le problème est impossible. Il est intéressant de noter que notre nouvel algorithme utilise une stratégie unifiée et générale pour chaque configuration initiale

Periodic Metro Scheduling

We introduce the { extsc{Periodic Metro Sched-ul-ing}} ({ extsc{PMS}}) problem, which aims in generating a periodic timetable for a given set of routes and a given time period, in such a way that the minimum time distance between any two successive trains that pass from the same point of the network is maximized. This can be particularly useful in cases where trains use the same rail segment quite often, as happens in metropolitan rail networks. We present exact algorithms for ({ extsc{PMS}}) in chain and spider networks, and constant ratio approximation algorithms for ring networks and for a special class of tree networks. Some of our algorithms are based on a reduction to the { extsc{Path Coloring}} problem, while others rely on techniques specially designed for the new problem

Network verification via routing table queries

The network verification problem is that of establishing the accuracy of a high-level description of its physical topology, by making as few measurements as possible on its nodes. This task can be formalized as an optimization problem that, given a graph and a query model specifying the information returned by a query at a node, asks for finding a minimum-size subset of nodes to be queried so as to univocally identify the graph. This problem has been studied with respect to different query models, assuming that a node had some global knowledge about the network. Here, we propose a new query model based on the local knowledge a node instead usually has. Quite naturally, we assume that a query at a given node returns the associated routing table, i.e., a set of entries which provides, for each destination node, a corresponding (set of) first-hop node(s) along an underlying shortest path