Shortest, Fastest, and Foremost Broadcast in Dynamic Networks

Highly dynamic networks rarely offer end-to-end connectivity at a given time. Yet, connectivity in these networks can be established over time and space, based on temporal analogues of multi-hop paths (also called {\em journeys}). Attempting to optimize the selection of the journeys in these networks naturally leads to the study of three cases: shortest (minimum hop), fastest (minimum duration), and foremost (earliest arrival) journeys. Efficient centralized algorithms exists to compute all cases, when the full knowledge of the network evolution is given. In this paper, we study the {\em distributed} counterparts of these problems, i.e. shortest, fastest, and foremost broadcast with termination detection (TDB), with minimal knowledge on the topology. We show that the feasibility of each of these problems requires distinct features on the evolution, through identifying three classes of dynamic graphs wherein the problems become gradually feasible: graphs in which the re-appearance of edges is {\em recurrent} (class R), {\em bounded-recurrent} (B), or {\em periodic} (P), together with specific knowledge that are respectively $n$ (the number of nodes), $\Delta$ (a bound on the recurrence time), and $p$ (the period). In these classes it is not required that all pairs of nodes get in contact -- only that the overall {\em footprint} of the graph is connected over time. Our results, together with the strict inclusion between $P$, $B$, and $R$, implies a feasibility order among the three variants of the problem, i.e. TDB[foremost] requires weaker assumptions on the topology dynamics than TDB[shortest], which itself requires less than TDB[fastest]. Reversely, these differences in feasibility imply that the computational powers of $R_n$, $B_\Delta$, and $P_p$ also form a strict hierarchy

Line-Recovery by Programmable Particles

Shape formation has been recently studied in distributed systems of programmable particles. In this paper we consider the shape recovery problem of restoring the shape when $f$ of the $n$ particles have crashed. We focus on the basic line shape, used as a tool for the construction of more complex configurations. We present a solution to the line recovery problem by the non-faulty anonymous particles; the solution works regardless of the initial distribution and number $f<n-4$ of faults, of the local orientations of the non-faulty entities, and of the number of non-faulty entities activated in each round (i.e., semi-synchronous adversarial scheduler)

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Symmetries and sense of direction in labeled graphs

AbstractWe consider edge-labeled graphs which model distributed systems, focus on properties of edge-labelings, and study their impact on graph classes. In particular, we investigate the relation between symmetries, topologies and sense of direction. We study symmetries based on the notion of view and of surrounding, and characterize the corresponding graph classes. Among other results, we show that the completely surrounding symmetric labeled graphs coincides with the class of Cayley graphs with Cayley labelings. We then focus on the relationship between symmetries and sense of direction in regular graphs. We characterize the class of regular labeled graphs with minimal symmetric sense of direction, as well as the class of those with group-based sense of direction