Computability of Perpetual Exploration in Highly Dynamic Rings

Abstract

We consider systems made of autonomous mobile robots evolving in highly dynamic discrete environment i.e., graphs where edges may appear and disappear unpredictably without any recurrence, stability, nor periodicity assumption. Robots are uniform (they execute the same algorithm), they are anonymous (they are devoid of any observable ID), they have no means allowing them to communicate together, they share no common sense of direction, and they have no global knowledge related to the size of the environment. However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location. A highly dynamic environment is modeled by a graph such that its topology keeps continuously changing over time. In this paper, we consider only dynamic graphs in which nodes are anonymous, each of them is infinitely often reachable from any other one, and such that its underlying graph (i.e., the static graph made of the same set of nodes and that includes all edges that are present at least once over time) forms a ring of arbitrary size.In this context, we consider the fundamental problem of perpetual exploration: each node is required to be infinitely often visited by a robot. This paper analyzes the computability of this problem in (fully) synchronous settings, i.e., we study the deterministic solvability of the problem with respect to the number of robots. We provide three algorithms and two impossibility results that characterize, for any ring size, the necessary and sufficient number of robots to perform perpetual exploration of highly dynamic rings.

Self-stabilizing Robots in Highly Dynamic Environments

“…In summary, previous work on exploration of dynamic graphs restricts strongly the dynamic of the considered graph. The notable exception is a recent work on perpetual exploration of highly dynamic rings [5]. This paper shows that, three (resp.…”

“…We give a first answer to this question by exhibiting the necessary and sufficient numbers of such robots to perpetually explore any connected-over-time ring, i.e., any dynamic ring with very weak assumption on connectivity: every node is infinitely often reachable from any another one without any recurrence, periodicity, nor stability assumption. More precisely, we showed that necessary and sufficient numbers of robots proved in [5] in a fault-free setting (2 robots for rings of size 3 and 3 robots for rings of size greater than 4) still hold in the self-stabilizing setting at the price of the loss of anonymity of robots. In addition to the above contributions, our results overcome the robot networks state-of-the-art in a couple of ways.…”

“…4) even if robots are not anonymous. Note that these necessity results are not implied by the ones of [5] (that focuses on anonymous robots). Then, Sections 4 and 5 present and prove two algorithms showing the sufficiency of these conditions.…”

“…First, we prove (see Theorem 3.1) that two robots with distinct identifiers are not able to perpetually explore in a self-stabilizing way connected-over-time rings of size greater than 4. Then, we show that we can borrow arguments from [5] to prove Theorem 3.2 that states that only one robot cannot complete the self-stabilizing perpetual exporation of connected-over-time rings of size greater than 3.…”

Gracefully Degrading Gathering in Dynamic Rings

“…On the other hand, few algorithms have been designed for robots evolving in dynamic graphs. The majority of them deals with the problem of exploration [3,4,11,13,18] (robots must visit each node of the graph at least once or infinitely often depending on the variant of the problem). In the most related work to ours [19], Di Luna et al study the gathering problem in dynamic rings.…”