In this paper, we introduce the notion of snapstabilization. A snap-stabilizing algorithm protocol guarantees that, starting from an arbitrary system configuration, the protocol always behaves according to its specification. So, a snap-stabilizing protocol is a self-stabilizing protocol which stabilizes in 0 steps.We propose a snap-stabilizing Propagation of Information with Feedback (PIF) scheme on a rooted tree network. We call this scheme Propagation of information with Feedback and Cleaning (P F C ). We present two algorithms. The first one is a basic P F C scheme which is inherently snapstabilizing. However, it can be delayed Oh 2 steps (where h is the height of the tree) due to some undesirable local states. The second algorithm improves the worst delay of the basic P F C algorithm from Oh 2 to 1 step. The P F C scheme can be used to implement the distributed reset, the distributed infimum computation, and the global synchronizer in O1 waves (or PIF cycles). Moreover, assuming that a (local) checking mechanism exists to detect transient failures or topological changes, the P F C scheme allows processors to (locally) "detect" if the system is stabilized, in O1 waves without using any global metric (such as the diameter or size of the network).Finally, we show that the state requirement for both P F C algorithms matches the exact lower bound of the PIF algorithms on tree networks-3 states per processor, except for the root and leaf processors which use only 2 states. Thus, the proposed algorithms are optimal PIF schemes in terms of the number of states.

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Two mobile agents starting at different nodes of an unknown network have to meet. This task is known in the literature as rendezvous. Each agent has a different label which is a positive integer known to it, but unknown to the other agent. Agents move in an asynchronous way: the speed of agents may vary and is controlled by an adversary. The cost of a rendezvous algorithm is the total number of edge traversals by both agents until their meeting. The only previous deterministic algorithm solving this problem has cost exponential in the size of the graph and in the larger label. In this paper we present a deterministic rendezvous algorithm with cost polynomial in the size of the graph and in the length of the smaller label. Hence we decrease the cost exponentially in the size of the graph and doubly exponentially in the labels of agents.As an application of our rendezvous algorithm we solve several fundamental problems involving teams of unknown size larger than 1 of labeled agents moving asynchronously in unknown networks. Among them are the following problems: team size, in which every agent has to find the total number of agents, leader election, in which all agents have to output the label of a single agent, perfect renaming in which all agents have to adopt new different labels from the set {1, . . . , k}, where k is the number of agents, and gossiping, in which each agent has initially a piece of information (value) and all agents have to output all the values. Using our rendezvous algorithm we solve all these problems at cost polynomial in the size of the graph and in the smallest length of all labels of participating agents.

The contribution of this paper is threefold. First, we present the paradigm of snap-stabilization. A snapstabilizing protocol guarantees that, starting from an arbitrary system configuration, the protocol always behaves according to its specification. So, a snap-stabilizing protocol is a time optimal self-stabilizing protocol (because it stabilizes in 0 rounds). Second, we propose a new Propagation of Information with Feedback (PIF) cycle, called Propagation of Information with Feedback and Cleaning (PFC). We show three different implementations of this new PIF. The first one is a basic PFC cycle which is inherently snap-stabilizing. However, the first PIF cycle can be delayed O(h 2 ) rounds (where h is the height of the tree) due to some undesirable local states. WARNING:The concept of snap-stabilization was first introduced in [12]. The concept evolved over the last eight years. We take this evolution in consideration in this paper, which includes the early results published in [10] and [12]. In particular, infinite repetition of computation cycles is a requirement of self -stabilizing systems. This is not required in snap-stabilization because snap-stabilization ensures that the first completed computation cycle is executed according to the specification of the problem. The correctness proofs conform to this basic property.The second algorithm improves the worst delay of the basic PFC algorithm from O(h 2 ) to 1 round. The state requirement for the above two algorithms is 3 states per processor, except for the root and leaf processors that use only 2 states. Also, they work on oriented trees. We then propose a third snap-stabilizing PIF algorithm on un-oriented tree networks. The state requirement of the third algorithm depends on the degree of the processors, and the delay is at most h rounds. Next, we analyze the maximum waiting time before a PIF cycle can be initiated whether the PIF cycle is infinitely and sequentially repeated or launch as an isolated PIF cycle. The analysis is made for both oriented and un-oriented trees. We show or conjecture that the two best of the above algorithms produce optimal waiting time. Finally, we compute the minimal number of states the processors require to implement a single PIF cycle, and show that both algorithms for oriented trees are also (in addition to being time optimal) optimal in terms of the number of states.

Leader election and arbitrary pattern formation are fundammental tasks for a set of autonomous mobile robots. The former consists in distinguishing a unique robot, called the leader. The latter aims in arranging the robots in the plane to form any given pattern. The solvability of both these tasks turns out to be necessary in order to achieve more complex tasks.In this paper, we study the relationship between these two tasks in a model, called CORDA, wherein the robots are weak in several aspects. In particular, they are fully asynchronous and they have no direct means of communication. They cannot remember any previous observation nor computation performed in any previous step. Such robots are said to be oblivious. The robots are also uniform and anonymous, i.e, they all have the same program using no global parameter (such that an identity) allowing to differentiate any of them. Moreover, none of them share any kind of common coordinate mechanism or common sense of direction, except that they agree on a common handedness (chirality).In such a system, Flochini et al. proved in [9] that it is possible to solve the leader election problem for n ≥ 3 robots if the arbitrary pattern formation is solvable for n ≥ 3. In this paper, we show that the converse is true for n ≥ 4 and thus, we deduce that both problems are equivalent for n ≥ 4 in CORDA provided the robots share the same chirality. The possible equivalence for n = 3 remains an open problem in CORDA.

In this article, we show that some fundamental self-and snap-stabilizing wave protocols (e.g., token circulation, PIF, etc.) implicitly assume a very light property that we call BreakingIn. We prove that BreakingIn is strictly induced by self-and snap-stabilization. Combined with a transformer, BreakingIn allows to easily turn the non-fault-tolerant versions of those protocols into snap-stabilizing versions. Unlike the previous solutions, the transformed protocols are very efficient and work at least with the same daemon as the initial versions extended to satisfy BreakingIn. Finally, we show how to use an additional property of the transformer to design snap-stabilizing extensions of those fundamental protocols like Mutual Exclusion.

Two mobile agents starting at different nodes of an unknown network have to meet. This task is known in the literature as rendezvous. Each agent has a different label which is a positive integer known to it, but unknown to the other agent. Agents move in an asynchronous way: the speed of agents may vary and is controlled by an adversary. The cost of a rendezvous algorithm is the total number of edge traversals by both agents until their meeting. The only previous deterministic algorithm solving this problem has cost exponential in the size of the graph and in the larger label. In this paper we present a deterministic rendezvous algorithm with cost polynomial in the size of the graph and in the length of the smaller label. Hence we decrease the cost exponentially in the size of the graph and doubly exponentially in the labels of agents.As an application of our rendezvous algorithm we solve several fundamental problems involving teams of unknown size larger than 1 of labeled agents moving asynchronously in unknown networks. Among them are the following problems: team size, in which every agent has to find the total number of agents, leader election, in which all agents have to output the label of a single agent, perfect renaming in which all agents have to adopt new different labels from the set {1, . . . , k}, where k is the number of agents, and gossiping, in which each agent has initially a piece of information (value) and all agents have to output all the values. Using our rendezvous algorithm we solve all these problems at cost polynomial in the size of the graph and in the smallest length of all labels of participating agents.

No abstract