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

Rendezvous of Heterogeneous Mobile Agents in Edge-weighted Networks

We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal times for the other agent. The goal of the agents is to meet on an edge or a node of the graph. In this scenario, we study the time required by the agents to meet, compared to the meeting time $T_{OPT}$ in the offline scenario in which the agents have complete knowledge about each others speed characteristics. When no additional assumptions are made, we show that rendezvous in our model can be achieved after time $O(n T_{OPT})$ in a $n$-node graph, and that such time is essentially in some cases the best possible. However, we prove that the rendezvous time can be reduced to $\Theta (T_{OPT})$ when the agents are allowed to exchange $\Theta(n)$ bits of information at the start of the rendezvous process. We then show that under some natural assumption about the traversal times of edges, the hardness of the heterogeneous rendezvous problem can be substantially decreased, both in terms of time required for rendezvous without communication, and the communication complexity of achieving rendezvous in time $\Theta (T_{OPT})$

A variant of the multi-agent rendezvous problem

The classical multi-agent rendezvous problem asks for a deterministic algorithm by which $n$ points scattered in a plane can move about at constant speed and merge at a single point, assuming each point can use only the locations of the others it sees when making decisions and that the visibility graph as a whole is connected. In time complexity analyses of such algorithms, only the number of rounds of computation required are usually considered, not the amount of computation done per round. In this paper, we consider $\Omega(n^2 \log n)$ points distributed independently and uniformly at random in a disc of radius $n$ and, assuming each point can not only see but also, in principle, communicate with others within unit distance, seek a randomised merging algorithm which asymptotically almost surely (a.a.s.) runs in time O(n), in other words in time linear in the radius of the disc rather than in the number of points. Under a precise set of assumptions concerning the communication capabilities of neighboring points, we describe an algorithm which a.a.s. runs in time O(n) provided the number of points is $o(n^3)$. Several questions are posed for future work.Comment: 18 pages, 3 figures. None of the authors has any previous experience in this area of research (multi-agent systems), hence we welcome any feedback from specialist

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations

Target Assignment in Robotic Networks: Distance Optimality Guarantees and Hierarchical Strategies

We study the problem of multi-robot target assignment to minimize the total distance traveled by the robots until they all reach an equal number of static targets. In the first half of the paper, we present a necessary and sufficient condition under which true distance optimality can be achieved for robots with limited communication and target-sensing ranges. Moreover, we provide an explicit, non-asymptotic formula for computing the number of robots needed to achieve distance optimality in terms of the robots' communication and target-sensing ranges with arbitrary guaranteed probabilities. The same bounds are also shown to be asymptotically tight. In the second half of the paper, we present suboptimal strategies for use when the number of robots cannot be chosen freely. Assuming first that all targets are known to all robots, we employ a hierarchical communication model in which robots communicate only with other robots in the same partitioned region. This hierarchical communication model leads to constant approximations of true distance-optimal solutions under mild assumptions. We then revisit the limited communication and sensing models. By combining simple rendezvous-based strategies with a hierarchical communication model, we obtain decentralized hierarchical strategies that achieve constant approximation ratios with respect to true distance optimality. Results of simulation show that the approximation ratio is as low as 1.4

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 approach in the plane: A deterministic polynomial algorithm

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other's range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let $\Delta$ and $l$ be the initial distance separating the agents and the length of the shortest label, respectively. Assuming that $\Delta$ and $l$ are unknown to both agents, does there exist a deterministic approach algorithm always working at a cost that is polynomial in $\Delta$ and $l$? In this paper, we provide a positive answer to the above question by designing such an algorithm