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We present a self-stabilizing algorithm for the (asynchronous) unison problem which achieves an efficient trade-off between time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous 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 a $n$-node network of diameter $D$ and assuming a period $B \geq 2D+2$, our algorithm only requires $O(\log B)$ bits per node to achieve full polynomiality as it stabilizes in at most $2D-2$ rounds and $O(\min(n^2B, n^3))$ moves. In particular and to the best of our knowledge, it is the first self-stabilizing unison for arbitrary anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Finally, we show that our solution allows to efficiently simulate synchronous self-stabilizing algorithms in an asynchronous environment. This provides a new state-of-the-art algorithm solving both the leader election and the spanning tree construction problem in any identified connected network which, to the best of our knowledge, beat all existing solutions of the literature., Comment: arXiv admin note: substantial text overlap with arXiv:2307.06635

This paper deals with the trade-off between time, workload, and versatility in self-stabilization, a general and lightweight fault-tolerant concept in distributed computing.In this context, we propose a transformer that provides an asynchronous silent self-stabilizing version Trans(AlgI) of any terminating synchronous algorithm AlgI. The transformed algorithm Trans(AlgI) works under the distributed unfair daemon and is efficient both in moves and rounds.Our transformer allows to easily obtain fully-polynomial silent self-stabilizing solutions that are also asymptotically optimal in rounds.We illustrate the efficiency and versatility of our transformer with several efficient (i.e., fully-polynomial) silent self-stabilizing instances solving major distributed computing problems, namely vertex coloring, Breadth-First Search (BFS) spanning tree construction, k-clustering, and leader election.

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.

Two mobile agents (robots) have to meet in an a priori unknown bounded terrain modeled as a polygon, possibly with polygonal obstacles. Agents are modeled as points, and each of them is equipped with a compass. Compasses of agents may be incoherent. Agents construct their routes, but the actual walk of each agent is decided by the adversary: the movement of the agent can be at arbitrary speed, the agent may sometimes stop or go back and forth, as long as the walk of the agent in each segment of its route is continuous, does not leave it and covers all of it. We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present, or not, (2) compasses of both agents agree, or not, (3) agents have or do not have a map of the terrain with their positions marked. The cost of a rendezvous algorithm is the worst-case sum of lengths of the agents' trajectories until their meeting. For each scenario we design a deterministic rendezvous algorithm and analyze its cost. We also prove lower bounds on the cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are tight.

A mobile robot represented by a point moving in the plane has to explore an unknown terrain with obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point p of the terrain explores (sees) all points q of the terrain for which the segment pq belongs to the terrain, and the limited vision, when we require additionally that the distance between p and q be at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm is measured by the length of the trajectory of the robot. For unlimited vision we show an exploration algorithm with complexity O(P + D?k), where P is the total perimeter of the terrain (including perimeters of obstacles), D is the diameter of the convex hull of the terrain, and k is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity O(P + A + ?Ak), where A is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains, if one of the parameters A or k is known, or for c-fat terrains, where c is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain T with obstacles is c-fat if R/r ? c, where R is the radius of the smallest disc containing T and r is the radius of the largest disc contained in T .) We also prove a matching lower bound ?(P + A + ?Ak) on the complexity of exploration for limited vision, even if the terrain is known to the robot.