Dijkstra’s token ring algorithm is a fundamental example of a self-stabilizing algorithm for solving mutual exclusion in an asynchronous distributed system arranged as a rooted directed ring. This paper studies the self-stabilization of this algorithm using an approach based on propositional satisfiability. We propose a logical modeling framework for the asynchronous executions of the algorithm that rigorously captures the state update rules, as well as the mechanisms for detecting convergence toward a legitimate configuration or, conversely, divergence through the existence of cycles between illegitimate configurations. Furthermore, we also optimize the efficiency and scalability of the analysis by introducing an offset-based symmetry-breaking technique applied to the initial configurations, thereby significantly reducing redundant explorations of equivalent execution scenarios. In addition, we extend the study to restricted daemon assumptions to assess open challenges.

A team of mobile agents, starting from distinct nodes of a network modeled as an undirected graph, have to meet at the same node and simultaneously declare that they all met. Agents execute the same algorithm, which they start when activated by an adversary or when an agent enter their initial node. While executing their algorithm, agents move from node to node by traversing edges of the network in synchronous rounds. Their perceptions and interactions are always strictly local: they have no visibility beyond their current node and can communicate only with agents occupying the same node. This task, known as gathering, is one of the most fundamental problems in distributed mobile systems. Over the past decades, numerous gathering algorithms have been designed, with a particular focus on minimizing their time complexity, i.e., the worst-case number of rounds between the start of the earliest agent and the completion of the task. To solve gathering deterministically, a common widespread assumption is that each agent initially has an integer ID, called label, only known to itself and that is distinct from those of all other agents. Labels play a crucial role in breaking possible symmetries, which, when left unresolved, may make gathering impossible. But must all labels be pairwise distinct to guarantee deterministic gathering?

We address the perpetual grid exploration (PGE) by a swarm of autonomous, asynchronous, myopic, and luminous robots. We first show that it is impossible for the robots to explore the grid regardless of their number and the number of colors they can take if their visibility range is one. We also show that PGE is impossible with three oblivious robots that have a visibility range of two hops. We then present three optimal algorithms solving the problem. The first algorithm uses four oblivious robots with a visibility range of two, but assumes they agree on a common chirality. For the two other algorithms, no common chirality is assumed. The former uses three robots that have a visibility range of two and a two-color light. The latter uses three oblivious robots under visibility range three.

A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows it to gain information on $G$ as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most $D$ to go back to node $s$. The upper bound $D$ is fixed as being equal to $(1+α)r$, where $r$ is the eccentricity of node $s$ (i.e., the maximum distance from $s$ to any other node) and $α$ is any positive real constant. This task has been introduced by Duncan et al. [ACM Trans. Algorithms 2006] and is known as \emph{distance-constrained exploration}. The \emph{penalty} of an exploration algorithm running in $G$ is the number of edge traversals made by the agent in excess of $|E|$. Panaite and Pelc [J. Algorithms 1999] gave an algorithm for solving exploration without any constraint on the moves that is guaranteed to work in every graph $G$ with a (small) penalty in $\mathcal{O}(|V|)$. Hence, a natural question is whether we could obtain a distance-constrained exploration algorithm with the same guarantee as well. In this paper, we provide a negative answer to this question. We also observe that an algorithm working in every graph $G$ with a linear penalty in $|V|$ cannot be obtained for the task of \emph{fuel-constrained exploration}, another variant studied in the literature. This solves an open problem posed by Duncan et al. [ACM Trans. Algorithms 2006] and shows a fundamental separation with the task of exploration without constraint on the moves.

We consider distributed systems subject to frequent topological changes. Specifically, we assume the network topology evolves as a dynamic graph in which, at any point in time, the temporal distance between any two processes is at most $$\varDelta $$ . Under a synchronous message-passing model where processes have unique identifiers and know both $$\varDelta $$ and an upper bound N on the number of processes n, we provide a distributed self-stabilizing mutual exclusion algorithm working in that class of dynamic graphs. Our solution stabilizes in $$\mathcal{O}(\varDelta .N)$$ rounds using bounded local memories. Moreover, it achieves optimal waiting time: once stabilized, the maximum delay before a process enters its critical section is at most $$n-1$$ rounds. Our algorithm is actually a composition of several self-stabilizing building blocks that respectively achieve Leader Election, Unison, and Ranking. We also provide original self-stabilizing solutions for the latter two problems; for the self-stabilizing leader election, we use a solution given by Altisen et al. (Theoretical Computer Science, 2023).

We present a self-stabilizing algorithm for the unison problem which is efficient in time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous asynchronous connected networks in which even local ports are unlabeled. It makes no assumption on the daemon and thus stabilizes under the weakest one: the distributed unfair daemon. In an n-node network of diameter D and assuming the knowledge B ≥ 2D+2, our algorithm only requires Θ(log(B)) bits per node and is fully polynomial as it stabilizes in at most 2D+2 rounds and O(min(n²B, n³)) moves. In particular, it is the first self-stabilizing unison for arbitrary asynchronous anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Furthermore, we show that our solution can be used to efficiently simulate synchronous self-stabilizing algorithms in asynchronous environments. For example, this simulation allows us to design a new state-of-the-art algorithm solving both the leader election and the BFS (Breadth-First Search) spanning tree construction in any identified connected network which, to the best of our knowledge, beats all existing solutions in the literature.

In this paper, we consider the exploration of an infinite grid by a swarm of fully-synchronous robots with weak capabilities: they are disoriented, opaque, do not communicate explicitely, have limited visibility, and cannot occupy the same position at the same time. Our first result shows that, in this context, minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to our exploration problem that is optimal w.r.t. both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. More precisely, the first algorithm solves the problem using eight oblivious robots under visibility range two (this visibility being optimal when considering oblivious robots), and the second algorithm solves the problem under visibility range one using six robots and two colors (which is optimal under this visibility range). Finally, we also tackle the optimality in terms of number of robots. According to the lower bound given in Bramas et al. (2020), we propose an algorithm working with a minimum number of robots (five) under visibility range one. This latter uses twelve colors and also guarantees that nodes are visited infinitely often. • The infinite grid exclusive exploration by synchronous oblivious robots is impossible under visibility range one whatever by their number. • The infinite grid exclusive exploration can be achieved by eight synchronous oblivious robots under the optimal visibility range two. • The infinite grid exclusive exploration can be achieved by six synchronous robots under visibility range one using only two colors. • The infinite grid exclusive exploration can be achieved by only five synchronous robots under visibility range one, yet using twelve colors.

The development of trustworthy distributed algorithms requires the verification of some key properties with respect to the formal specification of the expected system executions. The atomic-state model (ASM) is the most commonly used computational model to reason on self-stabilizing algorithms. In this work, we propose methods and tools to automatically verify the self-stabilization of distributed algorithms defined in that model. To that goal, we exploit the similarities between the ASM and computational models issued from the synchronous programming area to reuse their associated verification tools, and in particular their model checkers. This allows the automatic verification of all safety properties (including bounded liveness) of any algorithm under various asynchrony assumptions (from fully asynchronous to fully synchronous) and regardless of the hypotheses on the network ( e.g. , on its topology, its edge and node labeling). • We propose a language-based framework to verify distributed algorithms written in the atomic-state model. • The approach is modular due to a clear separation between the description of algorithms, daemons, topologies, and properties. • We illustrate our proposal by verifying various self-stabilizing algorithms, solving both static and dynamic tasks. • The versatility does not come at the price of sacrificing too much efficiency in terms of verification time.

We consider identified directed networks where processes know an upper bound on the maximum ancestor distance. Under these settings, we study the conditions on the network topology allowing the self-stabilization of two fundamental problems: the leader election and the synchronous unison. We show that those two problems can be self-stabilizingly solved in our settings if and only if the network contains a unique source component. In particular, to show that our condition is sufficient, we propose two algorithms and study their complexity. Notice that our topological condition covers a wide spectrum of digraphs since, for example, strongly connected digraphs, dipaths, and out-trees have a unique source component.

Distributed systems are ubiquitous, and their distributed nature make them particularly vulnerable to faults. Being able to automatically recover from these faults is of utmost importance, and self-stabilization is a general and lightweight approach to tackle this problem. However, fully asynchronous self-stabilizing algorithms (FASS) are notoriously difficult to design and prove. It thus makes sense to create and prove a transformer that turns synchronous algorithms into FASSes.

Self-stabilization is a general paradigm that characterizes the ability of a distributed system to recover from transient faults. Since its introduction by Dijkstra in 1974, self-stabilization has been successfully applied to efficiently solve many networking tasks. However, most of the literature focuses on bidirectional networks. Now, in today's networks such as WSNs, some communication channels may be one-way only. Considering such network topologies, a.k.a. directed graphs, makes self-stabilization more complicated, and sometimes even impossible. In this paper, we investigate the gap in terms of requirements and efficiency when considering a directed graph instead of an undirected one as network topology for a self-stabilizing algorithm. Our case study is a variant of a synchronous unison algorithm proposed by Arora et al.; the synchronous unison being a clock synchronization problem.

A proof assistant is an appropriate tool to write sound proofs. The need of such tools in distributed computing grows over the years due to the scientific progress that leads algorithmic designers to consider always more difficult problems. In that spirit, the PADEC Coq library has been developed to certify self-stabilizing algorithms. Efficiency of self-stabilizing algorithms is mainly evaluated by comparing their stabilization times in rounds, the time unit that is primarily used in the self-stabilizing area. In this paper, we introduce the notion of rounds in the PADEC library together with several formal tools to help the certification of the complexity analysis of self-stabilizing algorithms. We validate our approach by certifying the stabilization time in rounds of the classical Dolev et al’s self-stabilizing Breadth-first Search spanning tree construction.

Abstract In this paper, we present sasa, an open-source SimulAtor of Self-stabilizing Algorithms. Self-stabilization defines the ability of a distributed algorithm to recover after transient failures. sasa is implemented as a faithful representation of the atomic-state model (also called the locally shared memory model with composite atomicity). This model is the most commonly used one in the self-stabilizing area to prove both the correct operation of self-stabilizing algorithms and complexity bounds on them. sasa encompasses all features necessary to debug, test and analyze self-stabilizing algorithms. All these facilities are programmable to enable users to accommodate to their particular needs. For example, asynchrony is modeled by programmable stochastic daemons playing the role of input sequence generators. Properties of algorithms can be checked using formal test oracles. The sasa distribution also provides several facilities to easily achieve (batch-mode) simulation campaigns. We show that the lightweight design of sasa allows to efficiently perform huge such campaigns. Following a modular approach, we have aimed at relying as much as possible the design of sasa on existing tools, including ocaml, dot and several tools developed in the Synchrone Group of the VERIMAG laboratory.

We consider self-stabilization and its weakened form called pseudo-stabilization. We study conditions under which (pseudo- and self-) stabilizing leader election is solvable in networks subject to frequent topological changes. To model such an high dynamics, we use the dynamic graph (DG) paradigm and study a taxonomy of nine important DG classes. Our results show that self-stabilizing leader election can only be achieved in the classes where all processes are sources. Furthermore, even pseudo-stabilizing leader election cannot be solved in all remaining classes, except in the class where at least one process is a timely source. We illustrate that result by proposing a pseudo-stabilizing leader election algorithm for the latter class. We also show that in this last case, the convergence time of pseudo-stabilizing leader election algorithms cannot be bounded. Nevertheless, we show that our solution is speculative since its convergence time can be bounded when the dynamics is not too erratic, precisely when all processes are timely sources.

Abstract We consider swarms of asynchronous oblivious robots evolving into an anonymous grid-shaped network. In this context, we investigate optimal (w.r.t. the number of robots) deterministic solutions for the terminating exploration problem. We first show lower bounds in the semi-synchronous model. Precisely, we show that at least three robots are required to explore any grid of at least three nodes, even in the probabilistic case. Then, we show that at least four (resp. five) robots are necessary to deterministically explore a $\bf(2,2)$-Grid (resp. a $\bf(3,3)$-Grid). We then propose deterministic algorithms in the asynchronous model. This latter being strictly weakest than the semi-synchronous model, all the aforementioned bounds still hold in that context. Our algorithms actually exhibit the optimal number of robots that is necessary to explore a given grid. Overall, our results show that except in two particular cases, three robots are necessary and sufficient to deterministically explore a grid of at least three nodes and then terminate. The optimal number of robots for the two remaining cases is four for the $\bf(2,2)$-Grid and five for the $\bf(3,3)$-Grid, respectively.

We initiate research on self-stabilization in highly dynamic identified message-passing systems where dynamics is modeled using time-varying graphs (TVGs). More precisely, we address the self-stabilizing leader election problem in three wide classes of TVGs: the class TCB (Δ) of TVGs with temporal diameter bounded by Δ, the class TCB (Δ) of TVGs with temporal diameter quasi-bounded by Δ, and the class TCR of TVGs with recurrent connectivity only, where TCB (Δ) ⊆ TCB (Δ) ⊆ TCR. We first study conditions under which our problem can be solved. Precisely, we introduce the notion of size-ambiguity to show that the assumption on the knowledge of the number n of processes is central. Our results reveal that, despite the existence of unique process identifiers, any deterministic self-stabilizing leader election algorithm working in the TVG class TCB (Δ) or TCR cannot be size-ambiguous, justifying why our solutions for those classes assume the exact knowledge of n. We then present three self-stabilizing leader election algorithms for the TVG classes TCB (Δ), TCB(Δ), and TCR, respectively.

We propose a self-stabilizing reset algorithm working in anonymous networks. This algorithm resets the network in a distributed non-centralized manner, as each process detecting an inconsistency may initiate a reset. It is also cooperative in the sense that it coordinates concurrent reset executions in order to gain efficiency. Our approach is general since our reset algorithm allows to build self-stabilizing solutions for various problems and settings. As a matter of fact, it applies to both static and dynamic specifications since we propose efficient self-stabilizing reset-based algorithms for the 1-minimal (f,g)-alliance (a generalization of the dominating set problem) in identified networks and the unison problem in anonymous networks. These two latter instantiations enhance the state of the art. Indeed, in the former case, our solution is more general than the previous ones; while in the latter case, the time complexity of the proposed unison algorithm is better than that of previous ones.

We study (deterministic) leader election in unidirectional rings of homonym processes that have no a priori knowledge on the number of processes. In this context, we show that there is no algorithm that solves process-terminating leader election for the class of asymmetric labeled rings. In particular, there is no process-terminating leader election algorithm in rings in which at least one label is unique. However, we show that process-terminating leader election is possible for the subclass of asymmetric rings, where multiplicity is bounded. We confirm this positive results by proposing two algorithms, which achieve the classical trade-off between time and space.

The IEEE802.15.4e standard for low power wireless sensor networks defines a new mode called Time Slotted Channel Hopping (TSCH) as Medium Access Control (MAC). TSCH allows highly efficient deterministic time-frequency schedules that are built and maintained by the 6TiSCH operation sublayer (6top). In this paper, we propose a solution to limit the allocation of identical cells to co-located pair of nodes by distributed TSCH scheduling algorithms. It consists of making nodes able to overhear past cell negotiations exchanged in shared cells by their neighbors and prevent the nodes from reusing already assigned cells in future allocations. Our mechanism has been tested through simulations that show a significant improvement with respect to random scheduling algorithms.

We propose a general framework to build certified proofs of distributed self-stabilizing algorithms with the proof assistant Coq. We first define in Coq the locally shared memory model with composite atomicity, the most commonly used model in the self-stabilizing area. We then validate our framework by certifying a non trivial part of an existing silent self-stabilizing algorithm which builds a $k$-clustering of the network. We also certify a quantitative property related to the output of this algorithm. Precisely, we show that the computed $k$-clustering contains at most $\lfloor \frac{n-1}{k+1} \rfloor + 1$ clusterheads, where $n$ is the number of nodes in the network. To obtain these results, we also developed a library which contains general tools related to potential functions and cardinality of sets.

We propose a framework to build certified proofs of self-stabilizing algorithms using the proof assistant Coq. We first define in Coq the locally shared memory model with composite atomicity , the most commonly used model in the self-stabilizing area. We then validate our framework by certifying a non-trivial part of an existing self-stabilizing algorithm which builds a k -hop dominating set of the network. We also certify a quantitative property related to its output: we show that the size of the computed k -hop dominating set is at most \(\lfloor \frac{n-1}{k+1} \rfloor + 1\) , where n is the number of nodes. To obtain these results, we developed a library which contains general tools related to potential functions and cardinality of sets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We deal with the problem of maintaining a shortest-path tree rooted at some process r in a network that may be disconnected after topological changes. The goal is then to maintain a shortest-path tree rooted at r in its connected component, V\_r, and make all processes of other components detecting that r is not part of their connected component. We propose, in the composite atomicity model, a silent self-stabilizing algorithm for this problem working in semi-anonymous networks, where edges have strictly positive weights. This algorithm does not require any a priori knowledge about global parameters of the network. We prove its correctness assuming the distributed unfair daemon, the most general daemon. Its stabilization time in rounds is at most 3nmax+D, where nmax is the maximum number of non-root processes in a connected component and D is the hop-diameter of V\_r. Furthermore, if we additionally assume that edge weights are positive integers, then it stabilizes in a polynomial number of steps: namely, we exhibit a bound in O(maxi nmax^3 n), where maxi is the maximum weight of an edge and n is the number of processes.

We propose SR3, a secure and resilient algorithm for convergecast routing in WSNs. SR3 uses lightweight cryptographic primitives to achieve data confidentiality and data packet unforgeability. SR3 has a security proven by formal tool. We made simulations to show the resiliency of SR3 against various scenarios, where we mixed selective forwarding, blackhole, wormhole, and Sybil attacks. We compared our solution to several routing algorithms of the literature. Our results show that the resiliency accomplished by SR3 is drastically better than the one achieved by those protocols, especially when the network is sparse. Moreover, unlike previous solutions, SR3 self-adapts after compromised nodes suddenly change their behavior.

In this paper, we propose a silent self-stabilizing asynchronous distributed algorithm for constructing a kclustering of any connected network with unique IDs. Our algorithm stabilizes in O(n) rounds, using O(log n) space per process, where n is the number of processes. In the general case, our algorithm constructs O(n/k) k-clusters. If the network is a Unit Disk Graph (UDG), then our algorithm is 7.2552k+O(1)competitive, that is, the number of k-clusters constructed by the algorithm is at most 7.2552k + O(1) times the minimum possible number of k-clusters in any k-clustering of the same network. More generally, if the network is an Approximate Disk Graph (ADG) with approximation ratio λ, then our algorithm is 7.2552λ2k + O(λ)-competitive. Our solution is based on the self-stabilizing construction of a data structure called the MIS Tree, a spanning tree of the network whose processes at even levels form a maximal independent set of the network. The MIS tree construction is the time bottleneck of our k-clustering algorithm, as it takes Θ(n) rounds in the worst case, while the rest of the algorithm takes O(D) rounds, where V is the diameter of the network. We would like to improve that time to be O(D), but we show that our distributed MIS tree construction is a P-complete problem.

In this paper, we propose two snap-stabilizing distributed algorithms for the committee coordination problem. In this problem, a committee consists of a set of processes and committee meetings are synchronized, so that each process participates in at most one committee meeting at a time. Snap-stabilization is a versatile technique allowing to design algorithms that efficiently tolerate transient faults. Indeed, after a finite number of such faults (e.g. memory corruptions, message losses, etc), a snap-stabilizing algorithm immediately operates correctly, without any external intervention. We design snap-stabilizing committee coordination algorithms enriched with some desirable properties related to concurrency, (weak) fairness, and a stronger synchronization mechanism called 2-Phase Discussion Time. From previous papers, we know that (1) in the general case, (weak) fairness cannot be achieved in the committee coordination, and (2) it becomes feasible provided that each process waits for meetings infinitely often. Nevertheless, we show that even under this latter assumption, it is impossible to implement a fair solution that allows maximal concurrency. Hence, we propose two orthogonal snap-stabilizing algorithms, each satisfying 2-phase discussion time, and either maximal concurrency or fairness. The algorithm implementing fairness requires that every process waits for meetings infinitely often. Moreover, for this algorithm, we introduce and evaluate a new efficiency criterion called the degree of fair concurrency. This criterion shows that even if it does not satisfy maximal concurrency, our snap-stabilizing fair algorithm still allows a high level of concurrency.

In this paper, our focus is to lower the communication complexity of self-stabilizing protocols below the need of checking every neighbor forever. Our contribution is threefold: (i) We provide new complexity measures for communication efficiency of self-stabilizing protocols, especially in the stabilized phase or when there are no faults, (ii) On the negative side, we show that for non-trivial problems such as coloring, maximal matching, and maximal independent set, it is impossible to get (deterministic or probabilistic) self-stabilizing solutions where every participant communicates with less than every neighbor in the stabilized phase, and (iii) On the positive side, we present protocols for maximal matching and maximal independent set such that a fraction of the participants communicates with exactly one neighbor in the stabilized phase.

A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the systems recovers from this catastrophic situation without external intervention in finite time. Uni-directional networks preclude many common techniques in self-stabilization from being used, such as preserving local predicates. In this paper, we investigate the intrinsic complexity of achieving self-stabilization in unidirectional anonymous general networks, and focus on the classical vertex coloring problem. Specifically, we prove a lower bound of n states per process (where n is the network size) and a recovery time of at least n(n-1)/2 actions in total. We also provide a deterministic algorithm with matching upper bounds that performs in arbitrary unidirectional anonymous graphs.

In this paper, we address the problem of k-out-of-lscr exclusion, a generalization of the mutual exclusion problem, in which there are lscr units of a shared resource, and any process can request up to k units (1 les k les lscr). We propose the first deterministic self-stabilizing distributed k-out-of-lscr exclusion protocol in message-passing systems for asynchronous oriented tree networks which assumes bounded local memory for each process.

In this paper, we address the deterministic rendezvous of mobile agents into any unoriented connected graph. The agents are autonomous, oblivious, move asynchronously. For this problem, we exhibit some time and space lower bounds as well as some necessary conditions. We also propose an algorithm that is space-optimal and asymptotically optimal in rounds.

Given an arbitrary network G of processes with unique IDs and no designated leader, and given a k-dominating set I C G, we propose a silent self-stabilizing distributed algorithm that computes a subset D of I which is a minimal k-dominating set of G. Using D as the set of cluster-heads, a partition of G into clusters, each of radius k, follows. The algorithm is comparison-based, requires O(log n) space per process, converges in O(n) rounds and O(n2) steps, where n is the size of the network, and works under an unfair scheduler.

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.

Self-stabilization is a strong property which guarantees that a network always resume a correct behavior starting from an arbitrary initial state. Weaker guarantees have later been introduced to cope with impossibility results: probabilistic stabilization only gives probabilistic convergence to a correct behavior. Also, weak-stabilization only gives the possibility of convergence. In this paper, we investigate the relative power of weak, self, and probabilistic stabilization, with respect to the set of problems that can be solved. We formally prove that in that sense, weak stabilization is strictly stronger that self-stabilization. Also, we refine previous results on weak stabilization to prove that, for practical schedule instances, a deterministic weak-stabilizing protocol can be turned into a probabilistic self-stabilizing one. This latter result hints at more practical use of weak-stabilization, as such algorithms are easier to design and prove than their (probabilistic) self-stabilizing counterparts.

In this announcement, we report recent results where we address the open problem of snap-stabilization in message-passing systems.

A snap-stabilizing protocol, starting from any arbitrary initial configuration, always behaves according to its specification. In this paper, we present the first snap-stabilizing depth-first search wave protocol for arbitrary rooted networks assuming an unfair daemon, i.e. assuming the weakest scheduling assumption (a preliminary version of this work was presented in OPODIS 2004, 8th International Conference on Principles of Distributed Systems, Grenoble (France)).