International audienceIn this paper, we study the exclusive perpetual exploration problem with mobile anonymous and oblivious robots in a discrete space. Our results hold for the most generic settings: robots are asynchronous and are not given any sense of direction, so the left and right sense (i.e. chirality) is decided by the adversary that schedules robots for execution, and may change between invocations of a particular robots (as robots are oblivious). We investigate both the minimal and the maximal number of robots that are necessary and sufficient to solve the exclusive perpetual exploration problem. On the minimal side, we prove that three deterministic robots are necessary and sufficient, provided that the size n of the ring is at least 10, and show that no protocol with three robots can exclusively perpetually explore a ring of size less than 10. On the maximal side, we prove that k = n − 5 robots are necessary and sufficient to exclusively perpetually explore a ring of size n when n is co-prime with k

International audienceIt follows from the definition of silent self-stabilization, and from the definition of proof-labeling scheme, that if there exists a silent self-stabilizing algorithm using ℓ-bit registers for solving a task T , then there exists a proof-labeling scheme for T using registers of at most ℓ bits. The first result in this paper is the converse to this statement. We show that if there exists a proof-labeling scheme for a task T , using ℓ-bit registers, then there exists a silent self-stabilizing algorithm using registers of at most O(ℓ + logn) bits for solving T , where n is the number of processes in the system. Therefore, as far as memory space is concerned, the design of silent self-stabilizing algorithms essentially boils down to the design of compact proof-labeling schemes. The second result in this paper addresses time complexity. We show that, for every task T with k-bits output size in n-node networks, there exists a silent self-stabilizing algorithm solving T in O(n) rounds, using registers of O(n 2 + kn) bits. Therefore, as far as running time is concerned, every task has a silent self-stabilizing algorithm converging in a linear number of rounds

Abstract. This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. All variants of this problem assume that any node can be simultaneously occupied by several searchers. This assumption may be unrealistic, e.g., in the case of searchers modeling physical searchers, or may require each individual node to provide additional resources, e.g., in the case of searchers modeling software agents. We thus investigate exclusive graph searching, in which no two or more searchers can occupy the same node at the same time, and, as for the classical variants of graph searching, we study the minimum number of searchers required to capture the intruder. This number is called the exclusive search number of the considered graph. Exclusive graph searching appears to be considerably more complex than classical graph searching, for at least two reasons: (1) it does not satisfy the monotonicity property, and (2) it is not closed under minor. Nevertheless, we design a polynomial-time algorithm which, given any tree T , computes the exclusive search number of T . Moreover, for any integer k, we provide a characterization of the trees T with exclusive search number at most k. This characterization allows us to describe a special type of exclusive search strategies, that can be executed in a distributed environment, i.e., in a framework in which the searchers are limited to cooperate in a distributed manner.

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This paper focuses on compact deterministic self-stabilizing solutions for the leader election problem. When the protocol is required to be silent (i.e., when communication content remains fixed from some point in time during any execution), there exists a lower bound of Ω(log n) bits of memory per node participating to the leader election (where n denotes the number of nodes in the system). This lower bound holds even in rings. We present a new deterministic (non-silent) self-stabilizing protocol for n-node rings that uses only O(log log n) memory bits per node, and stabilizes in O(n log 2 n) rounds. Our protocol has several attractive features that make it suitable for practical purposes. First, the communication model fits with the model used by existing compilers for real networks. Second, the size of the ring (or any upper bound on this size) needs not to be known by any node. Third, the node identifiers can be of various sizes. Finally, no synchrony assumption, besides a weakly fair scheduler, is assumed. Therefore, our result shows that, perhaps surprisingly, trading silence for exponential improvement in term of memory space does not come at a high cost regarding stabilization time or minimal assumptions.

International audienceWe tackle a practical version of the well known {\it graph searching} problem, where a team of robots aims at capturing an intruder in a graph. The robots and the intruder move along the edges of the graph. The intruder is invisible, arbitrary fast, and omniscient. It is caught whenever it stands on a node occupied by a robot, and cannot escape to a neighboring node. We study graph searching in the CORDA model of mobile computing: robots are asynchronous, and they perform cycles of {\it Look-Compute-Move} actions. Moreover, motivated by physical constraints, we consider the \emph{exclusive} property, stating that no two or more robots can occupy the same node at the same time. In addition, we assume that the network and the robots are anonymous. Finally, robots are \emph{oblivious}, i.e., each robot performs its move actions based only on its current ''vision'' of the positions of the other robots. Our objective is to characterize, for a graph $G$, the set of integers $k$ such that graph searching can be achieved by a team of $k$ robots starting from \emph{any} $k$ distinct nodes in $G$. Our main result consists in a full characterization of this set, for any asymmetric tree. Towards providing a characterization for all trees, including trees with non-trivial automorphisms, we have also provides a set of positive and negative results, including a full characterization for any line. All our positive results are based on the design of algorithms enabling \emph{perpetual} graph searching to be achieved with the desired number of robots

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