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

Self-Stabilizing Robots in Highly Dynamic Environments

International audienceThis paper deals with the classical problem of exploring a ring by a cohort of synchronous robots. We focus on the perpetual version of this problem in which it is required that each node of the ring is visited by a robot infinitely often.The challenge in this paper is twofold. First, we assume that the robots evolve in a highly dynamic ring, i.e., edges may appear and disappear unpredictably without any recurrence nor periodicity assumption. The only assumption we made is that each node is infinitely often reachable from any other node. Second, we aim at providing a self-stabilizing algorithm to the robots, i.e., the algorithm must guarantee an eventual correct behavior regardless of the initial state and positions of the robots. Our main contribution is to show that this problem is deterministically solvable in this harsh environment by providing a self-stabilizing algorithm for three robots

Anonymous Graph Exploration with Binoculars

International audienceWe investigate the exploration of networks by a mobile agent. It is long known that, without global information about the graph, it is not possible to make the agent halts after the exploration except if the graph is a tree. We therefore endow the agent with binoculars, a sensing device that can show the local structure of the environment at a constant distance of the agent current location.We show that, with binoculars, it is possible to explore and halt in a large class of non-tree networks. We give a complete characterization of the class of networks that can be explored using binoculars using standard notions of discrete topology. This class is much larger than the class of trees: it contains in particular chordal graphs, plane triangulations and triangulations of the projective plane. Our characterization is constructive, we present an Exploration algorithm that is universal; this algorithm explores any network explorable with binoculars, and never halts in non-explorable networks

Patrolling on Dynamic Ring Networks

We study the problem of patrolling the nodes of a network collaboratively by a team of mobile agents, such that each node of the network is visited by at least one agent once in every $I(n)$ time units, with the objective of minimizing the idle time $I(n)$. While patrolling has been studied previously for static networks, we investigate the problem on dynamic networks with a fixed set of nodes, but dynamic edges. In particular, we consider 1-interval-connected ring networks and provide various patrolling algorithms for such networks, for $k=2$ or $k>2$ agents. We also show almost matching lower bounds that hold even for the best starting configurations. Thus, our algorithms achieve close to optimal idle time. Further, we show a clear separation in terms of idle time, for agents that have prior knowledge of the dynamic networks compared to agents that do not have such knowledge. This paper provides the first known results for collaborative patrolling on dynamic graphs

More efficient periodic traversal in anonymous undirected graphs

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in minimisation of the length of the exploration period. This problem is unsolvable if the local port numbers are set arbitrarily. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. described an algorithm for assigning port numbers, and an oblivious agent (i.e. agent with no memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was previously proved to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most 4 1/3 n for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Moreover, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case

Almost optimal asynchronous rendezvous in infinite multidimensional grids

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O( ( d)), where r = min(r1, r2) and for r ≥ 1. r)δpolylog ( d r

The temporal explorer who returns to the base.

In this paper we study the problem of exploring a temporal graph (i.e. a graph that changes over time), in the fundamental case where the underlying static graph is a star on n vertices. The aim of the exploration problem in a temporal star is to find a temporal walk which starts at the center of the star, visits all leaves, and eventually returns back to the center. We present here a systematic study of the computational complexity of this problem, depending on the number k of time-labels that every edge is allowed to have; that is, on the number k of time points where each edge can be present in the graph. To do so, we distinguish between the decision version STAREXP(k) , asking whether a complete exploration of the instance exists, and the maximization version MAXSTAREXP(k) of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We fully characterize MAXSTAREXP(k) and show a dichotomy in terms of its complexity: on one hand, we show that for both k=2 and k=3 , it can be efficiently solved in O(nlogn) time; on the other hand, we show that it is APX-complete, for every k≥4 (does not admit a PTAS, unless P = NP, but admits a polynomial-time 1.582-approximation algorithm). We also partially characterize STAREXP(k) in terms of complexity: we show that it can be efficiently solved in O(nlogn) time for k∈{2,3} (as a corollary of the solution to MAXSTAREXP(k) , for k∈{2,3} ), but is NP-complete, for every k≥6

Live Exploration with Mobile Robots in a Dynamic Ring, Revisited

The graph exploration problem requires a group of mobile robots, initially placed arbitrarily on the nodes of a graph, to work collaboratively to explore the graph such that each node is eventually visited by at least one robot. One important requirement of exploration is the {\em termination} condition, i.e., the robots must know that exploration is completed. The problem of live exploration of a dynamic ring using mobile robots was recently introduced in [Di Luna et al., ICDCS 2016]. In it, they proposed multiple algorithms to solve exploration in fully synchronous and semi-synchronous settings with various guarantees when $2$ robots were involved. They also provided guarantees that with certain assumptions, exploration of the ring using two robots was impossible. An important question left open was how the presence of $3$ robots would affect the results. In this paper, we try to settle this question in a fully synchronous setting and also show how to extend our results to a semi-synchronous setting. In particular, we present algorithms for exploration with explicit termination using $3$ robots in conjunction with either (i) unique IDs of the robots and edge crossing detection capability (i.e., two robots moving in opposite directions through an edge in the same round can detect each other), or (ii) access to randomness. The time complexity of our deterministic algorithm is asymptotically optimal. We also provide complementary impossibility results showing that there does not exist any explicit termination algorithm for $2$ robots. The theoretical analysis and comprehensive simulations of our algorithm show the effectiveness and efficiency of the algorithm in dynamic rings. We also present an algorithm to achieve exploration with partial termination using $3$ robots in the semi-synchronous setting.Comment: 13 page

Ping Pong in dangerous graphs: Optimal black hole search with pure tokens

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